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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Let $F$ and $G$ be linear transformations. Find the transformation matrix in respect to basis $B$ and $C$. (polynomials)

Let $F$ and $G$ be two linear transformations that maps from $P_2(\mathbb{R})$ to $P_3(\mathbb{R})$, such that: $$F(p(t)) = tp(t)-p(1)\\G(p(t)) = (t-1)p(t)$$ Find the transformation matrices of $F$ ...
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1answer
8 views

Infinity norm greater $1$ implies that spectral radius greater $1$?

Suppose I have an arbitrary real matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ such that the sum of absolute values in each row is greater than $1$: $$\sum_{j=1}^n |A_{ij}|>1,\quad \forall ...
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3answers
42 views

Rank of $ A^2 +A + I$

Let $A$ be $6 \times 6$ real symmetric matrix of rank $5$. Find the rank of $A^2 +A +I$. Well i dont know any tool that can solve this question. The book says answer is $5$ but it could be wrong ...
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23 views

Showing that something is the zero matrix

I'm trying to prove that if the linear system of equations $Ax=b$, where $A$ is a matrix with $m$ rows and $n$ cloumns, only has a solution when $b=0$ then $A$ is the zero-matrix. My idea goes like ...
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1answer
19 views

How to prove $A( v \times w)$ = $\det A\cdot(Av \times Aw)$ , where $A\in \text {Mat}_{3 \times 3}(\mathbb R)$ is an orthogonal matrix.

How to prove $A( v \times w)$ = $\det A\cdot(Av \times Aw)$, where $A\in \text {Mat}_{3 \times 3}(\mathbb R)$ is an orthogonal matrix ($\det A = \pm 1$) and $v,w\in \mathbb R^3$. I've tried writing ...
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3answers
28 views

Alternative matrix representation for translation

The ''usual" way to write translation for $\textbf{v}\in \mathbb{R}^2$ is with the following $3\times3$ matrix $$ \left( {\begin{array}{ccc} 1 & 0 & x_{0} \\ 0 & 1 & y_{0} \\ ...
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1answer
24 views

Proving a linear map is injective

There is a linear map $T:V\rightarrow W$, and $U\subseteq V$ is a subspace such that $$U\cap \ker T=\{0_V\}.$$ I want to prove that the map $T'=T\big|_U:U\rightarrow W$ is also injective. I want to ...
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12 views

Formula for connect line between points based on less value of x-axis

I'm stuck to find an equation or formula that connect a straight line to a given n points, based on a less value of x-axis. For example, i have 5 points each has (x,y), want the equation to connect ...
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2answers
14 views

Prove that the Conjuguate gradient method converges in at most $n$ iterations

I am trying to probe this corollary in a numerical PDE book: If $A\in \mathbb{R^{n\times n}}$ is symmetric and positive definite, then for some index $m\leq n$ , the residual $r_m$ generated by the ...
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2answers
17 views

Sign of Composition of Permutations

Let $\sigma$ be a permutation in $S_5$ with a sign of $-1$. Let $\pi$ be any other arbitrary permutation of $\{1,...,5\}$. What is the sign of the composition $\pi^{-1} \sigma \pi$? Is there any ...
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Decomposing change in estimates

In this scenario, an estimate of a variable is given by the observed data multiplied by a representative weight. Let the observed data at the current time be $d_t$ and the weight $w_t$. The difference ...
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14 views

Isomorphisms with invariant linearly independent dense subset.

If $T$ is an isomorphism acting on a separable Banach space $X$, can we find a countable, dense, linearly independent set $D\subset X$ such that $T(D)=D$? If $X$ is finite dimensional, then the answer ...
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29 views

Help converting currencies of 3 countries (linear algebra problem)

Three countries, $A$, $B$ and $C$ trade goods and services in a closed economy. The percentage of the total production of each country which is consumed by any given country is given in the ...
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16 views

self-adjoint operator over a three dimensional vector space [on hold]

How do I prove that a self-adjoint operator over a three dimensional vector space, is a matrix $$X= \left( \begin{matrix} a & x\\ x^t & B \\ \end{matrix} \right),$$ ...
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4answers
43 views

$a-c = \frac{b}{2},\ a-b = \frac{c}{6},\ b+c = 32$ find $a$ =?

I am getting frustrated as I am fighting with this. Please help. I know $$ 2a = b+2c \\ 6a = 6b+c $$ but after this i get confuse what to do next ?
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33 views

number of linear maps from $V\to V$

Question is to find total number of linear maps from vector space $\mathbb R^3(\mathbb R)$ to vector space $\mathbb R (\mathbb R)$ which are not Onto? I think trivial map, $T (x, y,z)=0$ could be one ...
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2answers
18 views

Find Point on the line segment (7/8) of the way connecting points P and Q

with P = (4,3,-4) and Q = (5,-4,3). My thinking is take the distance between the two, which is (1,-7,7) and taking 7/8 of it which is (-7/8,-49/8,49/8). But I feel like that is wrong and I have to ...
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1answer
13 views

Form of Matrix for Reflection about a Line

I've seen a bunch of variations on the wonderful properties of this specific matrix. My textbook gives one algebraic form in particular that I'm having a bit of trouble verifying: Any help here? I ...
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0answers
17 views

Calculating Normals across a sphere with a wave-like vertex shader

This is a bit of a CS question, but more than not it's a 3D math problem. I've been trying to get the correct normals for a sphere I'm messing with using a vertex shader. The algorithm can be boiled ...
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1answer
20 views

product of Matrices notation meaning

I am trying to workout what the meaning of the notation is below. $D$ is a matrix and it is the product from $1$ to $n$. However, the $k(i)$ notation of the matrix and the subsequent $k:(1,2,....n) ...
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2answers
31 views

Prove that vector space and dual space have same dimension

As an exercise in my textbook, I need to prove that if $V$ is a finite dimensional vector space with dual space $V^*$ over $\mathbb{R}$, then dim$(V)$=dim$(V^*)$. Let $\omega\in V^*$ and let ...
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2answers
11 views

show that [T]β is a diagonal matrix

$V = P_1(R), T(a + b(x)) = (6a - 6b) + (12a - 11b)x$, and $β = \{3+4x, 2+3x\}$ Show that $[T]β$ is a diagonal matrix I am totally confused about how to write down the matrix form of this ...
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1answer
26 views

help with this demonstration of linear algebra [on hold]

V be a vector space over K of finite dimension, T: V* → W a linear transformation. Prove that there exists a unique v∈V such that: T(f)=f(v) for all f∈V*
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SVD of a parametrized matrix.

Suppose we have a parametrized matrix $Z(λ)\in R^{m\times n}$ where $λ\in(a,b)$ and $Ζ(λ)$ is an analytic function of $λ$, e.g. $Z(λ)=λA+(1-λ)B$ where $A,B \in R^{m\times n}$. In general, the ...
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1answer
33 views

Proof help - Linear Transformation [on hold]

$T:V\to W$ is a linear transformation, and $\dim(T\circ T(V))=\dim(T(V))$. Proof $T\circ T(V)=T(V)$.
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Maximizing inner product

Suppose we have two row vectors $a$ and $b$ of nonnegative real numbers such that, for $j<k$ $a_j\leq a_k$ and $b_j\leq b_k$. Let P be a permutation matrix. Can we prove (or disprove) that $$ ...
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1answer
31 views

For which values ${(a,1,0),(1,a,1), (0,1,a)}$ form a basis?

To the set be a basis we should have: $$x(a,1,0)+y(1,a,1)+z(0,1,a) = (0,0,0)\implies x=y=z=0$$ so: $$ax + y = 0\\x + ay + z = 0\\y + az = 0$$ which is a system that only has a unique solution if ...
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0answers
29 views

Is it possible to construct a system of equations for which the set of solutions is a plane?

I may just be overthinking it at this point, but I've struggled to come up with one system of equations that results in a plane. Is it possible? I can't seem to convince myself of a disproof that its ...
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4answers
90 views

Show that a given matrix always has an eigenvector in $\mathbb{R}$ Can somebody give a hint?

The given exercise is, for all $\theta$ in $\mathbb{R}$, show that the matrix always has an eigenvector in $\mathbb{R^2}$ $$ A = \begin{pmatrix} \cos(\theta) & \sin(\theta) \\ \sin(\theta) & ...
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2answers
398 views

Can the null space be empty?

I was reading a proof of the theorem that the range of a linear map $T$ is always a subspace of the target space, and when the author was showing that the $0$ vector was included in the range, he made ...
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2answers
22 views

simple systems of equations problem

Choose h and k such that the system has 1) no solution, 2) a unique solution, and 3) many solutions. Give separate answers for each part. x-3y=1, 2x+hy=k For 1) and 3), isn't that impossible? And ...
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0answers
22 views

W subspace of $\mathbb{R^4}$ with $\mbox{dim}W = 3$ and $U =\cdots $. Find $\mbox{dim} U+W$ and $\mbox{dim} U\cap W$

Let $W$ be a subspace of $\mathbb{R^4}$ with $\mbox{dim}W = 3$ and $U = \mbox{span}((1,2,1,3),(3,1,-1,4))$. Find $\mbox{dim} U+W$ and $\mbox{dim} U\cap W$ Well, so $\mbox{dim } U = 2$, clearly, ...
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1answer
12 views

An Optimal Value of a Diagonal Matrix $\Xi$ in $ H = U \Xi$

We have access to very accurate estimates of matrices $H$ and $U$ (both are $n \times k$, $n > k$) such that the following relationship holds $$ H = U \Xi$$ where $\Xi$ is a $k \times k$ diagonal ...
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32 views

Compute the Frobenius norm

I'm trying to compute the Frobenius norm of $L^{-1}$ where $$L^{-1}= I + N + N^2 + ... + N^{n-1}$$ and $N$ is strictly lower triangular. Can anyone suggest some way to do this?
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30 views

Group Action and “nice” Approximation!

Studying the action of a group $G$ on a set $X$ is naturally the same as looking at the group homomorphism $\alpha: G \rightarrow Perm(X)$. So, for a given group $G$, classifying all sets $X$, on ...
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3answers
26 views

Question about linear operator [on hold]

Let $S: R^2 \rightarrow R^2 $ be a linear operator such that $ S^2 = S$ and $ S\not= 0, S \not= I$. Prove that exists a ordered basis $B$ such that $ [S]_B = \left( \begin{array}{cc} 1 & 0 \\ 0 ...
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0answers
34 views

Does someone know about this problm? [on hold]

Find all linear transformations $$T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 , \qquad (x,y,z)\mapsto (u,v)$$ which map the plane in $\mathbb{R}^3$ given by $$x+y+z=1$$ onto the line in $\mathbb{R}^2$ ...
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1answer
31 views

Solving an underdetermined system of equations

I've had a little break in solving systems of equations and so I wanted to verify here that my own answer to this problem is 100% correct and done :) So I have the following problem: Solve the ...
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2answers
49 views

Matrices representing the same linear transformation

Suppose $T:V\rightarrow V$ is a linear operator. Let a basis for $V$ be $B_1=\{e_1,e_2,\cdots,e_n\}$ . Let $A$ be the matrix of $T$ relative to this basis. This means : $T(e_k) = \sum_{i=1}^n ...
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How can I scale the covariance matrix which represent a gaussian distribution ? [on hold]

I have a model genrated by using GMM the output is the mean and covariance matrix .I need to scale the cov matrix .for example I want to double the elipse that represent this gaussian .
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Show that $G$ acts faithfully on $S$

Let $G$ be the group of $2*2$ matrices with determinant $1$ over the four-element field $F$. Let $S$ be the set of lines through the origin in $F^2$. Show that $G$ acts faithfully on $S$. (The action ...
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1answer
15 views

Properties of the inverse of unit (lower) triangular matrix

Is there any special properties about the inverse of a unit lower triangular matrix? I'm trying to prove this: $$L^{-1}=I_n + N + N^2 + ... + N^{n-1}$$ where $L$ is a unit lower triangular matrix ...
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1answer
37 views

Can anybody provide some steps on how to do this? [on hold]

How do I find a basis for the given plane $x+y+z=1$ in $\mathbb{R}^3$?
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1answer
32 views

Tensor Algebra: Symmetrization & Antisymmetrization

Problem Given the tensor algebra: $$TV:=\sum_{k=0}^\infty{\bigotimes}^k V$$ Regard the symmetrization and antisymmetrization: ...
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1answer
31 views

Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.

Let $GL_2(Z_m)$ denote the multiplicative group of invertible $2 * 2$ matrices over the ring of integers modulo m. Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.
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1answer
14 views

Nonnegative solution to underdetermined linear system

I would like to show that the underdetermined system $Ax=b,\; x\ge 0$, with $b$ being a positive vector and $A$ being a binary matrix, has at least one solution. I've seen several other related ...
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1answer
17 views

Solving a system of xor equations?

How can I solve the following system of xor equations? k0 ⊕ k2 ⊕ k3 = 0011 k0 ⊕ k2 ⊕ k4 = 1010 k0 ⊕ k1 ⊕ k2 ⊕ k3 = 0110 How can I solve this system to know the ...
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1answer
51 views

what does $p(-1) = 0$ mean?

In a linear algebra problem, it asks me to determine the subespace spanned by $$ \left\{ p(x) \in \mathbb{R}^3 : p(-1) = 0 \right\}. $$ What does it mean?
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1answer
23 views

In $\mathbb{R}^{3}$, does an orthogonal basis of integer vectors exist such that none of their coordinates is $0$?

In $\mathbb{R}^{3}$, does an orthogonal basis $\{$ $(a_{1}, a_{2}, a_{3}),$ $(b_{1}, b_{2}, b_{3}),$ $(c_{1}, c_{2}, c_{3})$ $\}$ exist such that all $a_{i}, b_{i}, c_{i}$ are integers $\neq 0$?
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0answers
25 views

Characteristic Polynomial Calculation

I have a problem in my homework in which I have to find the characteristic polynomial of the following matrix: I know the final solution is: However, my answer keeps getting wrong whenever I ...