Vector Space

Problem 1

Montrer que $(E=\mathbb{R}_{+}^{} \times \mathbb{R}, T, )$ est un espace vectoriel réel avec $T(x,y)=(xx',y+y')$ et $*(x,y)=(x^k,ky)$ .

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Problem 2

Find a basis for the subspace $H \subset \mathbb{P}_{3}$ where $p(2)=0$ and $p^{\prime}(-1)=0$ . Answer: $\{\square\}$ .

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Problem 3

Prove that for any positive integer $n$ , if $H_{1}, H_{2}$ are subspaces of $\mathbb{R}^{n}$ , then $H_{1} \cap H_{2}$ is also a subspace.

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Problem 4

Prove one of these: 1) $H_{1} \cap H_{2}$ is a subspace of $\mathbb{R}^{n}$ . 2) Set $\{\vec{v}_{1}, \ldots, \vec{v}_{p}\}$ is linearly independent.

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Problem 5

Determine if the vector $u = \begin{bmatrix}-2 \\ -9 \\ -3 \\ 4\end{bmatrix}$ is in the subspace generated by $v_{1}, v_{2}, v_{3}$ .

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Problem 6

Prove one of these: 1) Range of $T$ equals column space of $A_{T}$ . 2) Exists $T$ with range $H$ for $p$ -dimensional $H$ .

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Problem 7

Find values of $t \in \mathbb{R}$ for which the subspace $W=\operatorname{lin}((1,2,1),(2,5,3),(1,3))$ is defined by one non-zero linear equation. Also, determine the equation for each value.

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Problem 8

Find a basis for the span of the vectors, a basis for its orthogonal complement, and all $c$ such that $[c, 1, 1, 0]^T$ is in the span.

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Problem 9

Determine if the subset $W$ of vectors in $\mathbb{R}^{3}$ with a 1 in the first component is a subspace of $V=\mathbb{R}^{3}$ .

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Problem 10

Determine if the set of symmetric $2 \times 2$ matrices, $T$ , is a vector subspace and find its smallest spanning set.

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Problem 11

Find bases for the vector spaces $V=\operatorname{lin}(v_{1}, v_{2})$ and $W=\operatorname{lin}(v_{3}, v_{4})$ .

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Problem 12

Find a basis for $\mathbb{R}^{3}$ where the vector $(1,2,3)$ has coordinates $3,1,2$ in that basis.

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Problem 13

Find all bases of the subspace $V$ in $\mathbb{R}^{4}$ defined by $x_{1}-x_{3}-x_{4}=0$ and $x_{2}-2x_{3}+x_{4}=0$ using vectors $v_{1}, v_{2}, v_{3}, v_{4}$ .

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Problem 14

Show if the set $w = \{(a, b, c) \mid a = b = c, (a, b, c) \in R^3\}$ is a subspace of $R^3$ .

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Problem 15

Is the set $w$ of all vectors of the form $(a, 0, 0)$ in $R^{\wedge} 3$ a subspace of $R^{\wedge} 3$ ?

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Problem 16

Find the basis and dimension of the subspace $V=\operatorname{lin}((1,2,1,1),(1,0,0,2),(1,4,2,0),(3,2,1,5))$ in $\mathbb{R}^{4}$ . Also, determine a system of linear equations whose solutions equal $V$ .

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Problem 17

Submit your name, surname, student number, group number, and problem number.
Problem 1: Given $V=\operatorname{lin}((1,2,1,1),(1,0,0,2),(1,4,2,0),(3,2,1,5))$ in $\mathbb{R}^{4}$ , find the basis, dimension, and a system of equations for $V$ .

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Problem 18

Let $V=\operatorname{lin}((1,2,1,1),(1,0,0,2),(1,4,2,0),(3,2,1,5))$ in $\mathbb{R}^{4}$ . Find basis, dimension of $V$ , and a linear equation system for $V$ .
For $W \subset \mathbb{R}^{5}$ given by $\begin{cases} x_{1}+x_{2}+2 x_{3}+2 x_{4}-x_{5}=0 \\ 2 x_{1}+3 x_{2}+x_{3}-x_{4}+x_{5}=0 \end{cases}$ find basis and dimension of $W$ .

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Problem 19

Find a basis $\mathcal{A}$ and the dimension of the subspace $V=\operatorname{lin}((2,1,3),(1,1,-1),(-1,-1,1),(6,4,4))$ in $\mathbb{R}^{3}$ .

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Problem 20

Prove that for any positive integer $n$ and $n \times n$ matrix $A$ with distinct real eigenvalues $\lambda_{1}, \ldots, \lambda_{p}$ , there is a $p$ -dimensional subspace $H$ of $\mathbb{R}^{n}$ invariant under $\vec{x} \mapsto A \vec{x}$ .

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Problem 21

Find a basis $\mathcal{A}$ and the dimension of the subspace $V=\operatorname{lin}((1,0,2,1),(1,1,0,2),(2,3,1,5),(0,1,-2,1))$ in $\mathbb{R}^{4}$ .

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Problem 22

Determine if the set $A=\{(x_{1}, x_{2}) \mid x_{1}, x_{2} \text{ are integers}\}$ is a subspace.

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Problem 23

Determine if the set $A=\{(x_{1}, x_{2}) \mid x_{1}, x_{2} \text{ are integers}\}$ is a subspace by checking vector addition and scalar multiplication.

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Problem 24

Prove that $R^{3}$ is the span of the vectors $(1,0,1)$ , $(2,0,3)$ and $(0,1,1)$ .

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Problem 25

Find a basis and dimension of the vector space $V=\operatorname{lin}((1,2,-6),(5,-2,3),(7,1,-4))$ . Also, find a homogeneous system of equations for $V$ and determine values of $t$ for which $\left(-t^{2}, 5,-2 t\right) \in V$ .

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Problem 26

Determine the dimension of the vector space $V$ formed by the vectors $(1,2,-6)$ , $(5,-2,3)$ , and $(7,1,-4)$ in $\mathbb{R}^{3}$ .

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Problem 27

Find the reduced row echelon form of the span of vectors $(1,2,-6)$ , $(5,-2,3)$ , $(7,1,-4)$ in $\mathbb{R}^{3}$ .

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Problem 28

Is $V=\mathbb{R}^{3}$ a vector space with addition $(a,b,c)+(x,y,z)=(a x,b y,c z)$ and scalar $r\cdot(a,b,c)=(0,0,0)$ ? A: Yes B: No

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Problem 29

In a vector space $V$ , which property may not hold: A, B, C, D, E, or F?

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Problem 30

Redefine matrix addition and scalar multiplication for $2 \times 2$ matrices. Which vector space axioms fail?

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Problem 31

Opisz przestrzeń $V=\operatorname{lin}((1,2,1,3),(2,5,2,7),(1,3,1,4)) \subset \mathbb{R}^{4}$ za pomocą równań liniowych.

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Problem 32

Jakie jest wymiary podprzestrzeni liniowej W, gdzie $W=\{(x, y) \in \mathbb{R}^{2}: x=y\}$ ? a. 0 b. 1 c. 2

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Problem 33

Is the set of vectors $\{a, b, c, d, e\}$ a vector space?

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Problem 34

True or False: The set of even integers is not a subspace of $\mathbb{R}$ under standard operations.

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Problem 35

Is the subset $H = \left\{A=\left(\begin{array}{cc}a+2 & a-2 \\ 0 & 0\end{array}\right), a \in \mathbb{R}\right\}$ a subspace of $V = M_{22}$ ?

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Problem 36

Is the set $H = \{A \in M_{22}: A=\begin{pmatrix}a & a-1 \\ 0 & 0\end{pmatrix}\}$ a subspace of the vector space $V=M_{22}$ ?

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Problem 37

Check if the subset $H$ of $2 \times 2$ matrices $M_{22}$ , given by $A = \left(\begin{array}{cc}a & a-1 \\ 0 & 0\end{array}\right)$ , is a subspace.

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Problem 38

Prove that the set $S=\left\{\left[\begin{array}{c}-s \\ s-5 t \\ 3 t+2 s\end{array}\right] \mid s, t \in \mathbb{R}\right\}$ is a subspace of $\mathbb{R}^{3}$ . Find spanning vectors and check linear independence.

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Problem 39

Find the orthogonal projection of $g(x)=x^{2}+x$ onto the even function subspace $E$ defined on $X=\{-1,0,1\}$ .

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Problem 40

Prove that for a linear transformation $f: V \rightarrow \mathbb{R}$ , there exists a unique vector $v \in V$ such that $f(u)=\langle u, v\rangle$ .

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Problem 41

Check if the vectors $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ -3 \end{pmatrix}$ span the vector space.

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Problem 42

¿El conjunto de vectores $\left\{\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\right\}$ genera un espacio vectorial?

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Problem 43

Check if the vectors $\begin{pmatrix}2 \\ 10\end{pmatrix}$ and $\begin{pmatrix}10 \\ 8\end{pmatrix}$ span $\mathbb{R}^2$ .

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Problem 44

Check if the vectors $\begin{pmatrix}0 \\ 5 \\ 1\end{pmatrix}$ , $\begin{pmatrix}0 \\ -1 \\ 3\end{pmatrix}$ , $\begin{pmatrix}-1 \\ -1 \\ 5\end{pmatrix}$ generate $\mathbb{R}^{3}$ .

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Problem 45

Check if the vectors $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ span $\mathbb{R}^2$ .

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Problem 46

Check if the vectors $\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 2 \end{pmatrix}$ span $R^{2}$ .

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Problem 47

Check if the vectors $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ , $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ , and $\begin{pmatrix} -1 \\ -2 \end{pmatrix}$ span $\mathbb{R}^2$ .

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Problem 48

Identify the Euclidean space $\mathbb{R}^{n}$ that is isomorphic to the vector space $\mathcal{P}_{8}$ .

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Problem 49

Identify the Euclidean space $\mathbb{R}^{n}$ that is isomorphic to the vector space $\mathbb{R}_{5 \times 3}$ . Options: $\mathbb{R}^{5}$ , $\mathbb{R}^{15}$ , $\mathbb{R}^{8}$ .

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Problem 50

Exercice 8 : Montrez que $F$ est un sous-espace de $\mathbb{R}^{3}$ , trouvez ses dimensions, et déterminez $F \cap G$ .
Exercice 9 : Donnez la base de $\mathbb{IR}_{3}[x]$ , trouvez une base et dimension de $F$ et $G$ , et examinez $F + G$ .

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Problem 51

Soit $E$ un $\mathrm{K}$ -espace vectoriel, $f$ et $g$ des endomorphismes. Montrer que $\operatorname{Im} f + \operatorname{Im} g$ et $\operatorname{Ker} f + \operatorname{Ker} g$ sont des sommes directes.

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Problem 52

Exercice 4 1- Soit $\mathrm{E}$ un espace vectoriel de dimension finie avec deux endomorphismes $\mathrm{f}$ et $\mathrm{g}$ . Montrer que si $E=\operatorname{Im} f + \operatorname{Im} g$ et $E=\operatorname{Ker} f + \operatorname{Ker} g$ , alors ces sommes sont directes. 2- Pour $f \in L(\mathbb{R}^{4})$ tel que $f^{2}=0$ , montrer que $\operatorname{rg}(f) \leq 2$ . Exercice 5 Trouver un endomorphisme de $\mathbb{R}^{3}$ dont le noyau est le sous-espace engendré par $u=(1,0,0)$ et $v=(1,1,1)$ .

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Problem 53

1. Trouvez une base et la dimension de l'ensemble $E_{1}=\{(x, y, z, t) \mid x+y+z+t=0, x-z-2t=0\}$ .
2. Vérifiez si les vecteurs $v_{1}=(1,1,1,1)$ , $v_{2}=(1,-1,1,0)$ , $v_{3}=(0,2,0,a)$ , $v_{4}=(2,0,2,1)$ forment une famille libre. Donnez une base et la dimension de $E_{2}=\operatorname{Vect}(v_{1}, v_{2}, v_{3}, v_{4})$ . Déterminez une représentation cartésienne de $E_{2}$ .
3. Pour $a \neq 1$ , trouvez une base et la dimension de $E_{1} \cap E_{2}$ , puis la dimension de $E_{1} + E_{2}$ .

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Problem 54

1) Show the first four Legendre polynomials form a basis for $\mathrm{P}_{3}$ . Find coordinates of $q(x)=14-6x+8x^{2}+42x^{3}$ relative to this basis. Determine polynomial $p(x)$ from basis and vector $\begin{bmatrix} 2 \\ -3 \\ 4 \\ 6 \end{bmatrix}_{B}$ .

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Problem 55

Find if the set $S=\left\{\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right],\left[\begin{array}{cc}5 & -6 \\ 7 & 8\end{array}\right],\left[\begin{array}{cc}-4 & -3 \\ 9 & -7\end{array}\right],\left[\begin{array}{cc}-1 & 5 \\ 8 & 7\end{array}\right]\right\}$ is linearly independent and if it forms a basis for 2x2 matrices.

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Problem 56

Show that the orthogonal complement $W^{\perp}$ is a subspace of $\mathbb{R}^{n}$ and find its basis if $W=\operatorname{span}\{e_{1}, e_{2}, e_{3}\} \subset \mathbb{R}^{5}$ .

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Problem 57

Find an orthogonal basis using Gram-Schmidt for the set $U = \{U_{1}, U_{2}, U_{3}\}$ in $\mathbb{R}_{2 \times 2}$ .

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Problem 58

True or false? (a) Every vector space has a zero vector. (b) A vector space can have multiple zero vectors. (c) If $a \mathbf{u}=b \mathbf{u}$ , then $a=b$ . (d) If $a \mathbf{u}=a \mathbf{v}$ , then $\mathbf{u}=\mathbf{v}$ .

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Problem 59

True or false? (a) If $W$ is a vector space in $V$ , then $W$ is a subspace of $V$ . (b) Is the empty set a subspace of every vector space? (c) Does $V$ contain a subspace $W \neq V$ ? (d) Is the intersection of any two subsets of $V$ a subspace? (e) Is any union of subspaces of $V$ a subspace?

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Problem 60

(a) The zero vector space has no basis. (b) Finite span implies a basis exists. (c) Not every vector space has a finite basis. (d) A vector space has at most one basis. (e) Finite basis means all bases have the same number of vectors. (f) For a finite dimensional space $V$ , if $S_{1}$ is independent and $S_{2}$ spans $V$ , then $|S_{1}| \leq |S_{2}|$ . (g) If $S$ spans $V$ , each vector in $V$ can be expressed uniquely as a linear combination of $S$ . (h) Every subspace of a finite dimensional space is finite dimensional. (i) An $n$ dimensional space $V$ has one 0-dimensional and one $n$ -dimensional subspace. (j) In an $n$ dimensional space $V$ , a set $S$ with $n$ vectors is independent iff it spans $V$ .

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Problem 61

Every subspace of $\mathbb{R}^{3}$ has infinite vectors. Is this true or false?

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Problem 62

If set $\mathrm{S}$ spans subspace $\mathrm{W}$ in vector space $V$ , can every vector in $\mathrm{W}$ be formed from $S$ ?
Select one: True False

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Problem 63

W przestrzeni $\mathbb{R}^{n}$ mamy bazy $v_{1}, \ldots, v_{n}$ i $w_{1}, \ldots, w_{n}$ . Udowodnij, że istnieje $w_{i}$ , dla którego oba układy są bazami $\mathbb{R}^{n}$ : $(w_{i}, v_{2}, \ldots, v_{n})$ i $(w_{1}, \ldots, w_{i-1}, v_{1}, w_{i+1}, \ldots, w_{n})$ .

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Problem 64

Find an orthogonal basis using Gram-Schmidt for $U=\{U_{1}, U_{2}, U_{3}\}$ in $\mathbb{R}_{2 \times 2}$ .

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Problem 65

Find an orthogonal basis using Gram-Schmidt for $U_1, U_2, U_3$ in $\mathbb{R}_{2 \times 2}$ under the Frobenius inner product.

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Problem 66

Find an orthogonal basis for subspace W using the Gram-Schmidt process on vectors: $\begin{bmatrix} 0 \\ 8 \\ 4 \end{bmatrix}$ , $\begin{bmatrix} 6 \\ 4 \\ -3 \end{bmatrix}$ .

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Problem 67

Check if these sets are subspaces: (a) Symmetric $n \times n$ matrices, (b) Singular $2 \times 2$ matrices.

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Problem 68

Entscheiden Sie, ob die folgenden Aussagen über Untervektorräume wahr sind: (a) $x+3y+5z=0$ in $\mathbb{R}^{3}$ , (b) $\operatorname{Re}(z_{1}) \cdot \operatorname{Im}(z_{2}) \geqq 0$ in $\mathbb{C}^{2}$ , (c) $f(x)=a e^{3x}+e$ in $\mathcal{C}^{1}(\mathbb{R})$ , (d) $\langle x | y \rangle=0$ für alle $y \in Y$ in $\mathbb{R}^{n}$ .

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Problem 69

Zadanie 1: Znajdź bazę ortonormalną przestrzeni $V=\{(x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{3}; x_{1}-2 x_{2}+2 x_{3}=0\}$ .

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Problem 70

Trouver la dimension des espaces vectoriels suivants : (a) polynômes de degré $\leq 7$ avec terme constant nul, (b) matrices diagonales $5 \times 5$ , (c) $\mathbb{R}^{2} \square$ , (d) matrices symétriques $3 \times 3$ , (e) polynômes de degré $\leq 3$ , (f) matrices antisymétriques $6 \times 6$ .

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Problem 71

Trouver une base $B$ pour l'espace des matrices triangulaires supérieures $2 \times 2$ .

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Problem 72

Trouver la dimension des espaces vectoriels suivants : (a) Polynômes de degré ≤ 4 avec $p(t)=-p(-t)$ (b) Polynômes de degré ≤ 5 avec terme constant nul (c) Polynômes de degré ≤ 2 tels que $\int_{0}^{3} p(t) d t=0$ (d) Polynômes de degré ≤ 4 avec $p^{\prime}(t)$ constant (e) Polynômes de degré ≤ 6 avec $p(2)=0$

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Problem 73

Trouvez une base $B$ pour l'espace vectoriel $V=\{\vec{v} \in \mathbb{R}^{4} \mid \vec{v}=<a, b, 2a+2b, 2a+4b>\}$ .

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Problem 74

Trouver une base $B$ pour l'espace vectoriel $V=\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb{M}_{2 \times 2} \mid 6a + 8d = 4b + 4c \}$ .

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Problem 75

Trouver une base $B$ pour les polynômes de la forme $p(x) = a x^{2} + b x + 2 a$ , avec $a, b \in \mathbb{R}$ .

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Problem 76

Trouver une base orthogonale $B$ de l'espace vectoriel $V = \{\vec{v} \in \mathbb{R}^{3} \mid 7a + b + 3c = 0\}$ .

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Problem 77

Trouver une base orthogonale $B$ de l'espace vectoriel $V=\{\vec{v}=\langle a, b, c\rangle \in \mathbb{R}^3 | 7a + b + 3c = 0\}$ .

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Problem 78

Trouver une base orthogonale $B$ de l'espace vectoriel $V=\{\vec{v}=<a, b, c>\in \mathbb{R}^{3} \mid 7a+b+3c=0\}$ .

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Problem 79

Calculez $f(0,0)$ et $f(3,-2)$ pour $f(x,y) = 5x + 9y$ . Montrez que $E = \{(x,y) \mid 5x + 9y = 0\}$ est un sous-espace de $\mathbb{R}^2$ .

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Problem 80

Finde eine Basis für den Untervektorraum $V=\{x \in \mathbb{R}^{4} \mid \langle x | w\rangle=0\}$ und prüfe, ob $v_1$ , $v_2$ , $v_3$ in $V$ liegen.

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Problem 81

Trouver une base orthogonale $B$ de l'espace vectoriel $V = \{\vec{v} = \langle a, b, c \rangle \mid 7a + 7b + 9c = 0\}$ .

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Problem 82

Trouver une base orthogonale $B$ de l'espace vectoriel $V=\{\vec{v}=\langle a, b, c\rangle \mid 7a+7b+9c=0\}$ .

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Problem 83

Trouve une base orthogonale pour le sous-espace vectoriel de $\mathbb{R}^{3}$ défini par $2x - 3y + 5z = 0$ .

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Problem 84

Trouvez une base de l'espace vectoriel défini par $\Delta: [x, y, z] = [1, -1, 2] + k[2, -2, 4], k \in \mathbb{R}$ .

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Problem 85

Trouvez une base de l'espace vectoriel défini par $\Delta: [x, y, z] = [1, -1, 2] + k[2, -2, 4], k \in \mathbb{R}$ .

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Problem 86

Trouvez une base de l'espace vectoriel défini par la droite $\Delta: [x, y, z] = [1, -1, 2] + k[2, -2, 4], k \in \mathbb{R}$ .

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Problem 87

Trouvez une base de l'espace vectoriel défini par $\Delta: [x, y, z] = [1, -1, 2] + k[2, -2, 4], k \in \mathbb{R}$ .

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Problem 88

Quels ensembles parmi A, B, C, D sont des sous-espaces vectoriels de $\mathbf{P}_{4}$ ? A: $p(t)=a t^{2}$ , B: $p(t)=a t^{4}$ , C: $p(t)=t^{4}+3 t$ , D: $p(t)=a t^{4}+3 t$ .

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Problem 89

Find orthonormal bases for the orthogonal complements $V^{\perp}$ and $W^{\perp}$ in $\mathbb{R}^{4}$ .

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Problem 90

Znajdź ortonormalne bazy przestrzeni $V=\operatorname{lin}((1,0,1,0),(0,1,0,2),(2,-2,2,-4))$ i $W=\{(x_{1}, x_{2}, x_{3}, x_{4}) \in \mathbb{R}^{4} \mid x_{1}+2 x_{2}-x_{3}+x_{4}=0\}$ .

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Problem 91

Find the orthogonal projection and symmetry formulas of $\mathbb{R}^{3}$ onto the plane defined by $x_{1}-x_{2}+2 x_{3}=0$ .

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Problem 92

Find orthonormal bases for the orthogonal complements $V^{\perp}$ and $W^{\perp}$ of the subspaces in $\mathbb{R}^{4}$ .

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Problem 93

Find an orthonormal basis for the orthogonal complement $V^{\perp}$ of the subspace $V$ spanned by $(1,0,1,0)$ and $(-1,1,0,2)$ .

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Problem 94

What is the range space of a $3 \times 4$ matrix $A \in M_{3 \times 4}(\mathbb{R})$ ? Choose from: a. $\mathbb{R}^{2}$ b. $\mathbb{R}$ c. None d. $\mathbb{R}^{4}$ e. $\mathbb{R}^{3}$

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Problem 95

Find an orthonormal basis for the orthogonal complement $V^{\perp}$ of the span of the vectors (1,0,1,0) and (-1,1,0,2).

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Problem 96

Find an orthonormal basis for the orthogonal complement $V^{\perp}$ of the subspace $V$ spanned by $(1,0,1,0)$ and $(-1,1,0,2)$ in $\mathbb{R}^{4}$ .

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Problem 97

Find a basis and dimension of the subspace $V=\operatorname{lin}((1,1,4),(2,1,6),(1,2,6))$ . For which $t$ does $v=(-1,1,t)$ belong to $V$ ?

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Problem 98

What is the dimension of the space of $3 \times 3$ real matrices, $M_{3}(\mathbb{R})$ ? a. 3 b. 6 c. 9 d. None e. 5

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Problem 99

Let $S=\{p(x)=a+bx+cx^2 \mid 5a-4b+c=0\}$ .
1. Prove $S$ is a subspace of $\mathbb{R}_{2}[x]$ .
2. Determine a basis for $S$ .
3. Calculate $\operatorname{dim}(S)$ .

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Problem 100

What is the dimension of the polynomial space $\mathbb{R}[x]$ ? a. 1 b. None c. finite d. $\infty$ e. 0

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1 2

Vector Space

Problem 1

Montrer que (E=R+∗×R,T,∗)(E=\mathbb{R}_{+}^{*} \times \mathbb{R}, T, *)(E=R+∗​×R,T,∗) est un espace vectoriel réel avec T(x,y)=(xx′,y+y′)T(x,y)=(xx',y+y')T(x,y)=(xx′,y+y′) et ∗(x,y)=(xk,ky)*(x,y)=(x^k,ky)∗(x,y)=(xk,ky).

Problem 2

Find a basis for the subspace H⊂P3H \subset \mathbb{P}_{3}H⊂P3​ where p(2)=0p(2)=0p(2)=0 and p′(−1)=0p^{\prime}(-1)=0p′(−1)=0. Answer: {□}\{\square\}{□}.

Problem 3

Prove that for any positive integer nnn, if H1,H2H_{1}, H_{2}H1​,H2​ are subspaces of Rn\mathbb{R}^{n}Rn, then H1∩H2H_{1} \cap H_{2}H1​∩H2​ is also a subspace.

Problem 4

Prove one of these: 1) H1∩H2H_{1} \cap H_{2}H1​∩H2​ is a subspace of Rn\mathbb{R}^{n}Rn. 2) Set {v⃗1,…,v⃗p}\{\vec{v}_{1}, \ldots, \vec{v}_{p}\}{v1​,…,vp​} is linearly independent.

Problem 5

Determine if the vector u=[−2−9−34]u = \begin{bmatrix}-2 \\ -9 \\ -3 \\ 4\end{bmatrix}u=⎣⎡​−2−9−34​⎦⎤​ is in the subspace generated by v1,v2,v3v_{1}, v_{2}, v_{3}v1​,v2​,v3​.

Problem 6

Prove one of these: 1) Range of TTT equals column space of ATA_{T}AT​. 2) Exists TTT with range HHH for ppp-dimensional HHH.

Problem 7

Find values of t∈Rt \in \mathbb{R}t∈R for which the subspace W=lin⁡((1,2,1),(2,5,3),(1,3))W=\operatorname{lin}((1,2,1),(2,5,3),(1,3))W=lin((1,2,1),(2,5,3),(1,3)) is defined by one non-zero linear equation. Also, determine the equation for each value.

Problem 8

Find a basis for the span of the vectors, a basis for its orthogonal complement, and all ccc such that [c,1,1,0]T[c, 1, 1, 0]^T[c,1,1,0]T is in the span.

Problem 9

Determine if the subset WWW of vectors in R3\mathbb{R}^{3}R3 with a 1 in the first component is a subspace of V=R3V=\mathbb{R}^{3}V=R3.

Problem 10

Determine if the set of symmetric 2×22 \times 22×2 matrices, TTT, is a vector subspace and find its smallest spanning set.

Problem 11

Find bases for the vector spaces V=lin⁡(v1,v2)V=\operatorname{lin}(v_{1}, v_{2})V=lin(v1​,v2​) and W=lin⁡(v3,v4)W=\operatorname{lin}(v_{3}, v_{4})W=lin(v3​,v4​).

Problem 12

Find a basis for R3\mathbb{R}^{3}R3 where the vector (1,2,3)(1,2,3)(1,2,3) has coordinates 3,1,23,1,23,1,2 in that basis.

Problem 13

Find all bases of the subspace VVV in R4\mathbb{R}^{4}R4 defined by x1−x3−x4=0x_{1}-x_{3}-x_{4}=0x1​−x3​−x4​=0 and x2−2x3+x4=0x_{2}-2x_{3}+x_{4}=0x2​−2x3​+x4​=0 using vectors v1,v2,v3,v4v_{1}, v_{2}, v_{3}, v_{4}v1​,v2​,v3​,v4​.

Problem 14

Show if the set w={(a,b,c)∣a=b=c,(a,b,c)∈R3}w = \{(a, b, c) \mid a = b = c, (a, b, c) \in R^3\}w={(a,b,c)∣a=b=c,(a,b,c)∈R3} is a subspace of R3R^3R3.

Problem 15

Is the set www of all vectors of the form (a,0,0)(a, 0, 0)(a,0,0) in R∧3R^{\wedge} 3R∧3 a subspace of R∧3R^{\wedge} 3R∧3?

Problem 16

Find the basis and dimension of the subspace V=lin⁡((1,2,1,1),(1,0,0,2),(1,4,2,0),(3,2,1,5))V=\operatorname{lin}((1,2,1,1),(1,0,0,2),(1,4,2,0),(3,2,1,5))V=lin((1,2,1,1),(1,0,0,2),(1,4,2,0),(3,2,1,5)) in R4\mathbb{R}^{4}R4. Also, determine a system of linear equations whose solutions equal VVV.

Problem 17

Problem 18

Problem 19

Find a basis A\mathcal{A}A and the dimension of the subspace V=lin⁡((2,1,3),(1,1,−1),(−1,−1,1),(6,4,4))V=\operatorname{lin}((2,1,3),(1,1,-1),(-1,-1,1),(6,4,4))V=lin((2,1,3),(1,1,−1),(−1,−1,1),(6,4,4)) in R3\mathbb{R}^{3}R3.

Problem 20

Prove that for any positive integer nnn and n×nn \times nn×n matrix AAA with distinct real eigenvalues λ1,…,λp\lambda_{1}, \ldots, \lambda_{p}λ1​,…,λp​, there is a ppp-dimensional subspace HHH of Rn\mathbb{R}^{n}Rn invariant under x⃗↦Ax⃗\vec{x} \mapsto A \vec{x}x↦Ax.

Problem 21

Find a basis A\mathcal{A}A and the dimension of the subspace V=lin⁡((1,0,2,1),(1,1,0,2),(2,3,1,5),(0,1,−2,1))V=\operatorname{lin}((1,0,2,1),(1,1,0,2),(2,3,1,5),(0,1,-2,1))V=lin((1,0,2,1),(1,1,0,2),(2,3,1,5),(0,1,−2,1)) in R4\mathbb{R}^{4}R4.

Problem 22

Determine if the set A={(x1,x2)∣x1,x2 are integers}A=\{(x_{1}, x_{2}) \mid x_{1}, x_{2} \text{ are integers}\}A={(x1​,x2​)∣x1​,x2​ are integers} is a subspace.

Problem 23

Determine if the set A={(x1,x2)∣x1,x2 are integers}A=\{(x_{1}, x_{2}) \mid x_{1}, x_{2} \text{ are integers}\}A={(x1​,x2​)∣x1​,x2​ are integers} is a subspace by checking vector addition and scalar multiplication.

Problem 24

Prove that R3R^{3}R3 is the span of the vectors (1,0,1)(1,0,1)(1,0,1), (2,0,3)(2,0,3)(2,0,3) and (0,1,1)(0,1,1)(0,1,1).

Problem 25

Problem 26

Determine the dimension of the vector space VVV formed by the vectors (1,2,−6)(1,2,-6)(1,2,−6), (5,−2,3)(5,-2,3)(5,−2,3), and (7,1,−4)(7,1,-4)(7,1,−4) in R3\mathbb{R}^{3}R3.

Problem 27

Find the reduced row echelon form of the span of vectors (1,2,−6)(1,2,-6)(1,2,−6), (5,−2,3)(5,-2,3)(5,−2,3), (7,1,−4)(7,1,-4)(7,1,−4) in R3\mathbb{R}^{3}R3.

Problem 28

Is V=R3V=\mathbb{R}^{3}V=R3 a vector space with addition (a,b,c)+(x,y,z)=(ax,by,cz)(a,b,c)+(x,y,z)=(a x,b y,c z)(a,b,c)+(x,y,z)=(ax,by,cz) and scalar r⋅(a,b,c)=(0,0,0)r\cdot(a,b,c)=(0,0,0)r⋅(a,b,c)=(0,0,0)? A: Yes B: No

Problem 29

In a vector space VVV, which property may not hold: A, B, C, D, E, or F?

Problem 30

Redefine matrix addition and scalar multiplication for 2×22 \times 22×2 matrices. Which vector space axioms fail?

Problem 31

Opisz przestrzeń V=lin⁡((1,2,1,3),(2,5,2,7),(1,3,1,4))⊂R4V=\operatorname{lin}((1,2,1,3),(2,5,2,7),(1,3,1,4)) \subset \mathbb{R}^{4}V=lin((1,2,1,3),(2,5,2,7),(1,3,1,4))⊂R4 za pomocą równań liniowych.

Problem 32

Jakie jest wymiary podprzestrzeni liniowej W, gdzie W={(x,y)∈R2:x=y}W=\{(x, y) \in \mathbb{R}^{2}: x=y\}W={(x,y)∈R2:x=y}? a. 0 b. 1 c. 2

Problem 33

Is the set of vectors {a,b,c,d,e}\{a, b, c, d, e\}{a,b,c,d,e} a vector space?

Problem 34

True or False: The set of even integers is not a subspace of R\mathbb{R}R under standard operations.

Problem 35

Is the subset H={A=(a+2a−200),a∈R}H = \left\{A=\left(\begin{array}{cc}a+2 & a-2 \\ 0 & 0\end{array}\right), a \in \mathbb{R}\right\}H={A=(a+20​a−20​),a∈R} a subspace of V=M22V = M_{22}V=M22​?

Problem 36

Is the set H={A∈M22:A=(aa−100)}H = \{A \in M_{22}: A=\begin{pmatrix}a & a-1 \\ 0 & 0\end{pmatrix}\}H={A∈M22​:A=(a0​a−10​)} a subspace of the vector space V=M22V=M_{22}V=M22​?

Problem 37

Check if the subset HHH of 2×22 \times 22×2 matrices M22M_{22}M22​, given by A=(aa−100)A = \left(\begin{array}{cc}a & a-1 \\ 0 & 0\end{array}\right)A=(a0​a−10​), is a subspace.

Problem 38

Problem 39

Find the orthogonal projection of g(x)=x2+xg(x)=x^{2}+xg(x)=x2+x onto the even function subspace EEE defined on X={−1,0,1}X=\{-1,0,1\}X={−1,0,1}.

Problem 40

Prove that for a linear transformation f:V→Rf: V \rightarrow \mathbb{R}f:V→R, there exists a unique vector v∈Vv \in Vv∈V such that f(u)=⟨u,v⟩f(u)=\langle u, v\ranglef(u)=⟨u,v⟩.

Problem 41

Check if the vectors (12)\begin{pmatrix} 1 \\ 2 \end{pmatrix}(12​) and (−1−3)\begin{pmatrix} -1 \\ -3 \end{pmatrix}(−1−3​) span the vector space.

Problem 42

Montrer que $(E=\mathbb{R}_{+}^{} \times \mathbb{R}, T, )$ est un espace vectoriel réel avec $T(x,y)=(xx',y+y')$ et $*(x,y)=(x^k,ky)$ .

Find a basis for the subspace $H \subset \mathbb{P}_{3}$ where $p(2)=0$ and $p^{\prime}(-1)=0$ . Answer: $\{\square\}$ .

Prove that for any positive integer $n$ , if $H_{1}, H_{2}$ are subspaces of $\mathbb{R}^{n}$ , then $H_{1} \cap H_{2}$ is also a subspace.

Prove one of these: 1) $H_{1} \cap H_{2}$ is a subspace of $\mathbb{R}^{n}$ . 2) Set $\{\vec{v}_{1}, \ldots, \vec{v}_{p}\}$ is linearly independent.

Determine if the vector $u = \begin{bmatrix}-2 \\ -9 \\ -3 \\ 4\end{bmatrix}$ is in the subspace generated by $v_{1}, v_{2}, v_{3}$ .

Prove one of these: 1) Range of $T$ equals column space of $A_{T}$ . 2) Exists $T$ with range $H$ for $p$ -dimensional $H$ .

Find values of $t \in \mathbb{R}$ for which the subspace $W=\operatorname{lin}((1,2,1),(2,5,3),(1,3))$ is defined by one non-zero linear equation. Also, determine the equation for each value.

Find a basis for the span of the vectors, a basis for its orthogonal complement, and all $c$ such that $[c, 1, 1, 0]^T$ is in the span.

Determine if the subset $W$ of vectors in $\mathbb{R}^{3}$ with a 1 in the first component is a subspace of $V=\mathbb{R}^{3}$ .

Determine if the set of symmetric $2 \times 2$ matrices, $T$ , is a vector subspace and find its smallest spanning set.

Find bases for the vector spaces $V=\operatorname{lin}(v_{1}, v_{2})$ and $W=\operatorname{lin}(v_{3}, v_{4})$ .

Find a basis for $\mathbb{R}^{3}$ where the vector $(1,2,3)$ has coordinates $3,1,2$ in that basis.

Find all bases of the subspace $V$ in $\mathbb{R}^{4}$ defined by $x_{1}-x_{3}-x_{4}=0$ and $x_{2}-2x_{3}+x_{4}=0$ using vectors $v_{1}, v_{2}, v_{3}, v_{4}$ .

Show if the set $w = \{(a, b, c) \mid a = b = c, (a, b, c) \in R^3\}$ is a subspace of $R^3$ .

Is the set $w$ of all vectors of the form $(a, 0, 0)$ in $R^{\wedge} 3$ a subspace of $R^{\wedge} 3$ ?

Find the basis and dimension of the subspace $V=\operatorname{lin}((1,2,1,1),(1,0,0,2),(1,4,2,0),(3,2,1,5))$ in $\mathbb{R}^{4}$ . Also, determine a system of linear equations whose solutions equal $V$ .

Find a basis $\mathcal{A}$ and the dimension of the subspace $V=\operatorname{lin}((2,1,3),(1,1,-1),(-1,-1,1),(6,4,4))$ in $\mathbb{R}^{3}$ .

Prove that for any positive integer $n$ and $n \times n$ matrix $A$ with distinct real eigenvalues $\lambda_{1}, \ldots, \lambda_{p}$ , there is a $p$ -dimensional subspace $H$ of $\mathbb{R}^{n}$ invariant under $\vec{x} \mapsto A \vec{x}$ .

Find a basis $\mathcal{A}$ and the dimension of the subspace $V=\operatorname{lin}((1,0,2,1),(1,1,0,2),(2,3,1,5),(0,1,-2,1))$ in $\mathbb{R}^{4}$ .

Determine if the set $A=\{(x_{1}, x_{2}) \mid x_{1}, x_{2} \text{ are integers}\}$ is a subspace.

Determine if the set $A=\{(x_{1}, x_{2}) \mid x_{1}, x_{2} \text{ are integers}\}$ is a subspace by checking vector addition and scalar multiplication.

Prove that $R^{3}$ is the span of the vectors $(1,0,1)$ , $(2,0,3)$ and $(0,1,1)$ .

Determine the dimension of the vector space $V$ formed by the vectors $(1,2,-6)$ , $(5,-2,3)$ , and $(7,1,-4)$ in $\mathbb{R}^{3}$ .

Find the reduced row echelon form of the span of vectors $(1,2,-6)$ , $(5,-2,3)$ , $(7,1,-4)$ in $\mathbb{R}^{3}$ .

Is $V=\mathbb{R}^{3}$ a vector space with addition $(a,b,c)+(x,y,z)=(a x,b y,c z)$ and scalar $r\cdot(a,b,c)=(0,0,0)$ ? A: Yes B: No

In a vector space $V$ , which property may not hold: A, B, C, D, E, or F?

Redefine matrix addition and scalar multiplication for $2 \times 2$ matrices. Which vector space axioms fail?

Opisz przestrzeń $V=\operatorname{lin}((1,2,1,3),(2,5,2,7),(1,3,1,4)) \subset \mathbb{R}^{4}$ za pomocą równań liniowych.

Jakie jest wymiary podprzestrzeni liniowej W, gdzie $W=\{(x, y) \in \mathbb{R}^{2}: x=y\}$ ? a. 0 b. 1 c. 2

Is the set of vectors $\{a, b, c, d, e\}$ a vector space?

True or False: The set of even integers is not a subspace of $\mathbb{R}$ under standard operations.

Is the subset $H = \left\{A=\left(\begin{array}{cc}a+2 & a-2 \\ 0 & 0\end{array}\right), a \in \mathbb{R}\right\}$ a subspace of $V = M_{22}$ ?

Is the set $H = \{A \in M_{22}: A=\begin{pmatrix}a & a-1 \\ 0 & 0\end{pmatrix}\}$ a subspace of the vector space $V=M_{22}$ ?

Check if the subset $H$ of $2 \times 2$ matrices $M_{22}$ , given by $A = \left(\begin{array}{cc}a & a-1 \\ 0 & 0\end{array}\right)$ , is a subspace.

Find the orthogonal projection of $g(x)=x^{2}+x$ onto the even function subspace $E$ defined on $X=\{-1,0,1\}$ .

Prove that for a linear transformation $f: V \rightarrow \mathbb{R}$ , there exists a unique vector $v \in V$ such that $f(u)=\langle u, v\rangle$ .

Check if the vectors $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ -3 \end{pmatrix}$ span the vector space.

Check if the vectors $\begin{pmatrix}2 \\ 10\end{pmatrix}$ and $\begin{pmatrix}10 \\ 8\end{pmatrix}$ span $\mathbb{R}^2$ .

Check if the vectors $\begin{pmatrix}0 \\ 5 \\ 1\end{pmatrix}$ , $\begin{pmatrix}0 \\ -1 \\ 3\end{pmatrix}$ , $\begin{pmatrix}-1 \\ -1 \\ 5\end{pmatrix}$ generate $\mathbb{R}^{3}$ .

Check if the vectors $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ span $\mathbb{R}^2$ .

Check if the vectors $\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 2 \end{pmatrix}$ span $R^{2}$ .

Check if the vectors $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ , $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ , and $\begin{pmatrix} -1 \\ -2 \end{pmatrix}$ span $\mathbb{R}^2$ .

Identify the Euclidean space $\mathbb{R}^{n}$ that is isomorphic to the vector space $\mathcal{P}_{8}$ .

Identify the Euclidean space $\mathbb{R}^{n}$ that is isomorphic to the vector space $\mathbb{R}_{5 \times 3}$ . Options: $\mathbb{R}^{5}$ , $\mathbb{R}^{15}$ , $\mathbb{R}^{8}$ .

Exercice 8 : Montrez que $F$ est un sous-espace de $\mathbb{R}^{3}$ , trouvez ses dimensions, et déterminez $F \cap G$ .
Exercice 9 : Donnez la base de $\mathbb{IR}_{3}[x]$ , trouvez une base et dimension de $F$ et $G$ , et examinez $F + G$ .

Soit $E$ un $\mathrm{K}$ -espace vectoriel, $f$ et $g$ des endomorphismes. Montrer que $\operatorname{Im} f + \operatorname{Im} g$ et $\operatorname{Ker} f + \operatorname{Ker} g$ sont des sommes directes.

Show that the orthogonal complement $W^{\perp}$ is a subspace of $\mathbb{R}^{n}$ and find its basis if $W=\operatorname{span}\{e_{1}, e_{2}, e_{3}\} \subset \mathbb{R}^{5}$ .

Find an orthogonal basis using Gram-Schmidt for the set $U = \{U_{1}, U_{2}, U_{3}\}$ in $\mathbb{R}_{2 \times 2}$ .

True or false? (a) Every vector space has a zero vector. (b) A vector space can have multiple zero vectors. (c) If $a \mathbf{u}=b \mathbf{u}$ , then $a=b$ . (d) If $a \mathbf{u}=a \mathbf{v}$ , then $\mathbf{u}=\mathbf{v}$ .

Every subspace of $\mathbb{R}^{3}$ has infinite vectors. Is this true or false?

If set $\mathrm{S}$ spans subspace $\mathrm{W}$ in vector space $V$ , can every vector in $\mathrm{W}$ be formed from $S$ ?
Select one: True False

Find an orthogonal basis using Gram-Schmidt for $U=\{U_{1}, U_{2}, U_{3}\}$ in $\mathbb{R}_{2 \times 2}$ .

Find an orthogonal basis using Gram-Schmidt for $U_1, U_2, U_3$ in $\mathbb{R}_{2 \times 2}$ under the Frobenius inner product.

Find an orthogonal basis for subspace W using the Gram-Schmidt process on vectors: $\begin{bmatrix} 0 \\ 8 \\ 4 \end{bmatrix}$ , $\begin{bmatrix} 6 \\ 4 \\ -3 \end{bmatrix}$ .

Check if these sets are subspaces: (a) Symmetric $n \times n$ matrices, (b) Singular $2 \times 2$ matrices.

Zadanie 1: Znajdź bazę ortonormalną przestrzeni $V=\{(x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{3}; x_{1}-2 x_{2}+2 x_{3}=0\}$ .

Trouver une base $B$ pour l'espace des matrices triangulaires supérieures $2 \times 2$ .

Trouvez une base $B$ pour l'espace vectoriel $V=\{\vec{v} \in \mathbb{R}^{4} \mid \vec{v}=<a, b, 2a+2b, 2a+4b>\}$ .

Trouver une base $B$ pour l'espace vectoriel $V=\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb{M}_{2 \times 2} \mid 6a + 8d = 4b + 4c \}$ .