Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Does a definite integral define a linear functional? [on hold]

Would $\displaystyle\int_0^1 t^2x(t)\,dt$ be a linear functional? For each $x$ in $P$ the function $y$ is defined by $\displaystyle\int_0^1 t^2x(t)\,dt$. I have to show that $y(ax+bz) = ay(x)+by(z)$. ...
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53 views

group action same thing as homomorphism

A linear group action of a group $G$ on a vector space $V$ is the same thing as a homomorphism from G to the general linear group $GL(V)$. attempt: Suppose a linear group action of a group $G$ on a ...
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Eigenvector of a matrix of ones associated with $\lambda =0$

An $n\times n$ matrix consistent of all ones, will have two eigenvalues: $0$ and $n$. The eigenvector associated with $n$ will be $(1,1,...,1)$, but are there then infinite solutions for the ...
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1answer
16 views

How to show $S_1\subset W_1$ and $S_2\subset W_2$ are independent $\implies$ $S_1\cup S_2$ is independent based on the following assumption?

Let $W_1$ and $W_2$ be subspaces of vector space $V$ satisfying $W_1\cap W_2=\{0\}$ ,how to show $S_1\subset W_1$ and $S_2\subset W_2$ are linearly independent $\implies$ $S_1\cup S_2$ is linearly ...
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Find the matrix representing T and Find the Image of T (as a span of vectors)

Let $T(a,b) = (a+b,2a-b,3a)$. a)Find the matrix representing $T$. b)Find the image of $T$ (as a span of vectors). So I found that $T$ is a linear transformation. Now would the matrix just be $A$= ...
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14 views

Basis for Vector Space iff can be Expressed Uniquely as Linear Combo of Basis

Let $V$ be a vector space and $\beta= \{ u_1,\dots ,u_n \}$ be a subset of $V$. $\Rightarrow$ $\beta$ is a basis for $V$ iff each vector $v\in V$ can be unquiley expressed as a linear combination of ...
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Is $S_1\cap S_2$ and $S_1\setminus S_2$ always linearly dependent if $S_1$ and $S_2$ are linearly dependent subsets of vector space $V$?

Let $S_1$ and $S_2$ be linearly dependent subsets of vector space $V$, are $S_1\cap S_2$ and $S_1\setminus S_2$ always linearly dependent? The counterexample for the first one I can think of is ...
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19 views

Computing a new inverse given information from an earlier inverse:

Suppose I have linearly independent set of vectors $v_1 ... v_k \in \mathbb{R}^{k}$. I can let $B = [v_1 ... v_k]$ and by applying Gaussian elimination on the matrix $$ [ B \ I_k] $$ I end up ...
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23 views

F is a vector space and U, V, and W are subspaces of F. Prove that $U\bigcup V\bigcup W$ is a subspace of F if and only if $U,V\subset W $.

The return is a given, but what about the other implication? We couldn't solve it in class. Sorry about the formatting, I hope it isn't an issue.
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systems of equations with variables

I have the following problem in my homework Suppose a, b, are two constant paramaters such that the system below is consistent for any values of f and g. What can you say about the numbers a ...
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1answer
27 views

How do I get a solution set equals to a sub space?

I've four vectors that spans the $\mathbb{R}^4$ sub-space $W_1$: $\alpha_1 = \{-1,0,1,2 \}$, $\alpha_2 = \{3,4,-2,5 \}$, $\alpha_3 = \{0,4,1,11 \}$, $\alpha_4 = \{1,4,0,9 \}$ And I'm asked to ...
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1answer
32 views

why does matlab give me a negative number?

I have the following problem A steel company has four different types of scrap metal (called Typ-1 to Typ-4) with the following compositions per unit of volume They need to determine the volumes ...
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1answer
26 views

Suppose V and W are finite-dimensional. Let $v \in V$. Let $E=\{T \in \mathscr{L}(V,W)\ |\ Tv=0\}.$ Show E is subspace of $\mathscr{L}(V,W)$

Suppose V and W are finite-dimensional. Let $v \in V$. Let $$E=\{T \in \mathscr{L}(V,W)\ |\ Tv=0\}.$$ a.) Show E is subspace of $\mathscr{L}(V,W)$ b.) Suppose that $v \neq 0$, what is dim E? Here is ...
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How to show $P_3(R)=W\oplus W_1$ and $P_3(R)=W\oplus W_2$ based on the following assumption?

Let $W=$Span$\{1, x\}$, $W_1=$Span$\{x^2, x^3\}$ and $W_2=$Span$\{1+x+x^2+x^3, 1+x+x^2-x^3\}$, how to show $P_3(R)=W\oplus W_1$ and $P_3(R)=W\oplus W_2$? $P_3(R)=W+ W_1$ because Span$\{1, ...
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27 views

Matrix differentiation proof of quadratic product $x^TAx$

would appreciate any hints with the proof for $x^TAx$ using index notation: Suppose $x$ is an $n$ x 1 vector, $A$ is an $n$ x $n$ matrix. $A$ does not depend on $x$, and also $\alpha = x^TAx$. Let ...
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57 views

Subspace of linear transforms from V to V

Suppose V is finite-dimensional and the $\mathscr{E}$ is a subspace of $\mathscr{L}(V)$ such the $ST\in \mathscr{E}$ and $TS \in \mathscr{E}$ for all $S \in \mathscr{L}(V)$ and all $T \in ...
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Find the mapping of inverse of linear transformation

Check if $L(p)(x)=(1+4x)p(x)+(x-x^2)p'(x)-(x^2+x^3)p''(x)$ is a linear transformation on space of polynomials $P_2(x)$ where $p(x)=ax^2+bx+c$. If yes, find its matrix in standard basis and find the ...
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1answer
18 views

Simplification of a product of three matrices

Define $$\mathbf{c}_t = \begin{bmatrix} x_{1t} \\ x_{2t} \\ \vdots \\ x_{Nt} \end{bmatrix}\in \mathbb{R}^N$$ where all entries are in $\mathbb{R}$, $t = 1, 2, \dots, p+1$. I am trying to simplify ...
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30 views

How to find $U+W$ and $U\cap W$ based on the following assumption?

Let $U=\{(x_1, x_2, x_3, x_4)\in R^4\mid x_1+ x_2=0, x_3+ x_4=0 \}$, $W=\{(x_1, x_2, x_3, x_4)\in R^4\mid x_1+ x_3=0, x_2+ x_4=0 \}$, how to find $U+W$ and $U\cap W$? I think $U\cap W=\{(x_1, x_2, ...
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How to define and make the dot product of two continuous matrix?

I was thinking recently that i always learn algebra with discret basis. But in case where the basis is continuous, how can i define a continuous matrix and when it is define how can i do the dot ...
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1answer
11 views

Solving simultaneous linear congruences for two unknowns

Find all pairs $(x,y)$ which solve $$ \left\{ \begin{align} 9x+20y&\equiv0\mod{29}\\ 16x+13y&\equiv0\mod{29} \end{align} \right. $$ So I have written this in the form ...
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1answer
29 views

Find the dimension and a basis of a subspace

Let $U$ is the set of all commuting matrices with matrix $A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}$. Prove ...
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15 views

How to show that $V=$Span$(S_2)$ if Span$(S_1)=V$ and that every vector in $S_1$ is in Span$(S_2)$?

Let $S_1$ and $S_2$ be subsets of a vector space $V$. Assume Span$(S_1)=V$ and that every vector in $S_1$ is in Span$(S_2)$, how to show that $V=$Span$(S_2)$ as well? In my opinion, to show ...
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Finding the general solution of a system of linear equations

so I've come across this question in preparation for an exam: Let $A$ be a $4\times 4$ matrix where $rank(A)=3$. The vectors $(1,2,0,-1),(0,2,1,1)$ are solutions to the system ...
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How to rotate a 3D object, using only local x-, y-, and z-rotations, so that it always faces a camera at the origin

I have been struggling with a difficult problem involving 3D rotations. I first came across this problem in a computer science context, but I've attempted to generalize it a bit before posting. (I ...
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1answer
15 views

How to show that $R^3$ is the direct sum of $W_1=$Span$(1,1,1)$ and $W_2=$Span$(\{1,0,0\}, \{1,1,0\})$?

How to show that $R^3$ is the direct sum of $W_1=$Span$(1,1,1)$ and $W_2=$Span$(\{1,0,0\}, \{1,1,0\})$? So we write it as $R^3=W_1+W_2$ because every $(x_1, x_2, x_3)\in R^3$ can be written as ...
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Linear maps of polynomials, the bases of the space and their corresponding matrix.

Suppose $T \in \mathrm{Hom}(\mathscr{P}_3(\mathbb{R}),\mathscr{P}_4(\mathbb{R}))$ is defined by: $$Tp(x)=(x^2p(x))',$$ for all $x \in \mathbb{R}$ and $S \in\mathrm{Hom} ...
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1answer
20 views

Transformations and Dependence

Hi, for these problems I generally get the gist of it. If you have some linearly dependent vectors $v_1, \ldots, v_m$ in $\mathbb{R}^n$ then when you transform those vectors $T(v_1), \dots, T(v_m)$ ...
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1answer
55 views

Theorems restricting the eigenvalue of a matrix

I have a square matrix $C$, whose entries I will denote by $c_{ij}$, and I would like to bound the magnitude of its eigenvalues. Each $c_{ij}$ is defined in terms of $s_{ij}$ and $S_j$ as follows: ...
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How to prove that $W_1\cap W_2\supset$ Span$(S_1\cap S_2)$ if $W_1=$ Span$(S_1)$ and $W_2=$ Span$(S_2)$ are subspaces of vector space?

In my opinion, let $v\in$ Span($S_1\cap S_2$) and therefore $v\in$ Span$(S_1)$ and $v\in$ Span$(S_2)$. Write $v=c_1z_1+...+ c_nz_n$ where $z_k\in S_1\cap S_2$ and $c_k\in R$. Here I am feeling I have ...
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1answer
20 views

Almost completing the argument: $p: L \to L$ a projector, then $im~ p \oplus \ker p = L$.

I had to show that if $p: L \to L$ is a projector, then $im ~p \oplus \ker p = L$. This was easy. Now I have to show that the matrix of $p$ is divided on four blocks where one of them is $r$ ...
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1answer
36 views

If commutativity of vector space is omitted, can we still use other axioms to prove the commutativity?

Here I am thinking of using $-(x+y)$ and show that it equals $-(y+x)$. $-(x+y)=-x-y$ by distributivity =$-x-y+0=...$ Here I don't know how to continue, could someone suggest?
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11 views

Derivative with respect to vectors related through a matrix

Consider a function $g: \mathbb{R}^r \to \mathbb{R} $ and two vectors $\mathbf{b} \in \mathbb{R}^r$ and $\mathbf{c} \in \mathbb{R}^m$ such that $\mathbf{c} = \mathbf{A}\mathbf{b}$. If I calculate the ...
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19 views

How to show that the following is satisfied for all vector space axiom?

Let $V=\{a_2x^2+a_1x+a_0|a_1, a_2, a_3\in \mathbb{R}, a_2\ne 0\}$ with operation defined by $$(a_2x^2+a_1x+a_0)+(b_2x^2+b_1x+b_0)=(a_2+b_2)x^2+(a_1+b_1)x+(a_0+b_0)$$ ...
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1answer
25 views

Check if two square matrices are similar.

Check if matrices $A= \begin{bmatrix} 1 & 1 & 5 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix}$ and $B=\begin{bmatrix} 1 & 7 ...
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Can we attach a space with discrete signal?

This question refers to the link https://en.wikipedia.org/wiki/Space_(mathematics) and https://en.wikipedia.org/wiki/Discrete-time_signal. My question is how can we associate a discrete signal with a ...
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1answer
19 views

Showing that $O$ is the only nilpotent matrix in $\langle A \rangle$ where $A$ is diagonalizable

I have the following task: Let $A\in \mathcal{M}_n(K)$ be a diagonalizable square matrix. Show using the spectral decomposition of $A$ that the only nilpotent matrix in $\langle A\rangle ...
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28 views

is there a closed form expression for the following matrix infinite series

Consider this infinite sum of matrices. Is there any closed form to express this sum? $S=B+ABA^T +A^2B({A^T})^2+A^3B({A^T})^3+...$ And B is diagonal. Thanks
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25 views

Prove that $u$ is orthogonal to $v-\operatorname{proj}u(v)$ for all vectors $u$ and $v$ in $\mathbb{R}^{n}$ where $u \neq 0$.

Here's where I'm at, not sure where to go from here. Two vectors are orthogonal if their dot product is $0$. Knowing that; $$u \cdot (v - \operatorname{proj} u(v)) = 0$$ $$u \cdot \left(v - \frac{u ...
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1answer
25 views

How to show that $M_{2\times 2}(\mathbb{R})=W_1\oplus W_2$ based on the following assumption?

Let the subspaces $W_1=\{\begin{pmatrix}a&b\\-b&a \end{pmatrix}|a, b\in \mathbb{R}\}$ and $W_2=\{\begin{pmatrix}c&d\\d&-c \end{pmatrix}|c, d\in \mathbb{R}\}$ of $M_{2\times ...
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1answer
14 views

If $x+y=(x_1y_1, …, x_ny_n)$ and $c\cdot '\ x=x^c_1, …, x^c_n$, how to show that with these two operation $V$ is a subspace?

Let $V=(R^+)^n=\{(x_1, ..., x_n)| x_i\in R^+$for each $i\}$. In $V$ define a vector sum operation $+'$ by $x+y=(x_1y_1, ..., x_ny_n)$ and scalar multiplication $\cdot '$ by $c\cdot '\ x=x^c_1, ..., ...
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2answers
24 views

How to show if the following subset $W$ is a subspace of a vector space $V$?

$1.$ $V=P_n(\mathbb{R}), $and $ W=\{p(x)\in P_n(\mathbb{R})\mid p(1)+p(2)+p(3)=0 \}$ $2.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R}) \mid A \text{ is not symmetric}\}$ ...
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1answer
47 views

An inequality on the rank of a block matrix

Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix $$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$ where $A \in \mathbb ...
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1answer
24 views

Inner product in a direct sum of a dimensional space

Supposer that $V = W_{1} \oplus W_{2}$, $f_{1}$ and $f_{2}$ are inner product at $W_{1}$ and $W_{2}$, respectively. Show that there is only one inner product $f$ in $V$ such that i) $W_{2} = ...
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2answers
39 views

showing projection is a linear operator

Show that the orthogonal projection is linear. Let $x_i=y_i+z_i$, where $x_i\in X$, $y_i\in Y$, $z_i\in Y^\perp$, and $\alpha,\beta$ be scalars. Then \begin{align}P(\alpha x_1+\beta ...
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0answers
33 views

Transformation matrix from principal angles and vectors

If I got it right, given two planes in $N$-dimensional space ($N\gg2$), their 2 principal angles ($\theta_1$, $\theta_2$) and 4 vectors ($\vec{a}_1$, $\vec{a}_2$, $\vec{b}_1$, $\vec{b}_2$) can be ...
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0answers
15 views

Orthogonal projections exercise

Let $V$ be a $n-$dimensional space with inner product and consider $W$ a subspace of $V$. If $E$ it's a projections with $Im E = W$ such that $|E\alpha| \leq |\alpha|$ $\forall \alpha \in V$ then $E$ ...
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1answer
33 views

Basis for 4th degree polynomials such that integral of $p(x)$ from $-1$ to $1$ equals $0$

Let $U= \{ p \in \mathscr P_4\mathbb{R} \ | \int_{-1}^1 p(x)dx=0\}$. a.) Find a basis for $U$. b.) Find a subspace $W$ of $\mathscr{P_4}(\mathbb{R})$ such that $\mathscr{P_4}(\mathbb{R})= U \oplus ...
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29 views

In matrix algebra, what's the name for the inverse operation of pre- or post- multiplication?

For example, in this typical equation: $$\mathbf{Mv}-\lambda \mathbf{v}=\mathbf{0}$$ (where $\mathbf{M}$ is a symmetric matrix, $\mathbf{v}$ is a vector, $\lambda$ is a scalar, and $\mathbf{0}$ is a ...
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2answers
26 views

Proving that $u$ and $v$ are linearly independent, given the independence of $T(u)$ and $T(v)$

Suppose that $T$ is a linear transformation and that $T(u)$ and $T(v)$ are linearly independent. Prove that $u$ and $v$ are linearly independent. I have no idea where to start in this case. Just need ...