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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Distributive property on trace norm

I hope this is not a trivial question, basically, if we have trace norm of $A$ defined as $||A||_\star := \operatorname{trace}\left(\sqrt{A^*A}\right) = \sum\limits_{i=1}^{\min\{m,n\}} \sigma_i$, if $...
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1answer
26 views

Find a linearly independent set of vectors that spans the same subspace of $\mathbb{R}^3$

Find a linearly independent set of vectors that spans the same subspace of $\mathbb{R}^3$ as that spanned by the vectors $\begin{bmatrix}2\\2\\-1\end{bmatrix}, \begin{bmatrix}-8\\-2\\5\end{bmatrix}, ...
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1answer
29 views

Determining a basis for Col($A$) and a dimension for the null space of $A$

Let $A = \begin{bmatrix}1&-1&1&0&-2&1\\1&-1&1&1&0&0\\-1&1&-1&2&5&-1\end{bmatrix}$ a) Determine a basis for Col($A$) b) What is the ...
3
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1answer
41 views

Lower bounding the trace of $A^2$ using the trace of $A^T A$

$\DeclareMathOperator{\tr}{tr}$For a real, square matrix $A$, I believe that one has a simple upper bound on the (absolute value of the) trace of its square in terms of the trace of its Gramian-type ...
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1answer
34 views

What if 1st pivot is missing but the 2nd one is there?

I have the following matrix : $$A= \begin{bmatrix} 0 &1 &2 &3 &4 \\ 0 &0 &0 &1 &2\\ 0 &0 &0 &0 &0\\ \end{bmatrix} $$ So here the 1st pivot is missing ...
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1answer
29 views

Find a basis of nullspace(A)

Let $A = \begin{bmatrix}4&-4&2&-6\\2&-2&1&-3\end{bmatrix}$ Find a basis of nullspace$(A)$ I first put $A$ in RREF to get: $\begin{bmatrix}1&-1&1/2&-3/2\\0&0&...
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15 views

Linearity of projection of angle

In the book Putnam and Beyond, problem 252 reads as follows: Consider the angle formed by two half-lines in three-dimensional space. Prove that the average of the measure of the projection of the ...
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2answers
30 views

Finding the matrix representation of a transformation

Question is : The vectors $(2,1)$ and $(1,1)$ form a basis for $R^2$. Let $T$ be a linear transformation satisfying $T(2,1)=(-2,6)$ and $T(1,1)=(0,5)$. Find the matrix of $T$ with respect to the ...
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4answers
58 views

Proof/Intuition for Eigenvalues to Solve Linear Differential Equations

To solve an equation of the form $$\frac{dx}{dt}=ax+by \\ \frac{dy}{dt}=cx+dy$$ Does anyone know the reasoning why you solve for eigenvalues and eigenvectors to determine the functions for x and y ...
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1answer
27 views

Is “Find a basis for the column space of $A$” and “Find a basis for Col $A$” asking the same thing?

Is "Find a basis for the column space of $A$" and "Find a basis for Col $A$" asking the same thing? Basically my question, it may seem stupid. I'm just curious though I'm trying to make sure I ...
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2answers
34 views

Is it possible to define the tensor product of two vectors with respect to a bilinear form?

Given two vectors $\vec{v},\vec{w} \in \mathbb{R}^n$, and a bilinear form $\mathcal{B}$ represented by an $n \times n$ matrix $B$, we can define the inner product of $\vec{v}$ and $\vec{w}$ with ...
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0answers
26 views

Using the Tensor Product Construction to Show Linear Independence of the Standard Basis

For simplicity's sake, let's assume $V = \mathbb{R}^2$ with standard basis $e_1, e_2$. I construct the tensor product $V \otimes V$ as the quotient of the the free vector space $F(V \times V)$ by the ...
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1answer
52 views

Does there exist a multiplicative $f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that $f\neq x\mapsto x^a$ for all $a$?

If we consider the functional equation: $f:\mathbb{Q}^+\to\mathbb{R}$ such that $$ f(xy)=f(x)f(y) $$ for all $x,y\in\mathbb{Q}^+$ I think, I have constructed a solution which is not of the form $x\...
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1answer
28 views

Prove that the size and the number of Jordan block's of $\lambda$ and $\bar{\lambda}$ are the same.

Prove that the size and the number of Jordan block's of $\lambda$ and $\bar{\lambda}$ are the same where $T$ is a real operator on $V$ a finite dimensional space. I know that the main is to show that ...
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1answer
50 views

Matrices that represent rotations

So the question is What 3 by 3 matrices represent the transformations that a) rotate the x-y plane, then x-z, then y-z through 90? I believe this is the matrix that rotates the xy plane \begin{...
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2answers
29 views

Calculate the spectral norm

Consider the four vectors $v_1, v_2, u_1, u_2 \in \mathbb{C}^2$ with $$v_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \qquad v_2 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \qquad u_1 = \begin{pmatrix} ...
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0answers
18 views

Positive values of Quadratic Form with nonnegative vectors

I hope somebody can give me a hint on the following problem: Consider a real valued $n \times n$ matrix $A$. Let $x$ be a real-valued "non-negative" vector, i.e. a vector with nonnegative components,...
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1answer
39 views

$x=x_p+x_n$ is given, asked to find the matrix

The question is : Find a 2 by 3 system $Ax=b$ whose complete solution is : $$ x=\begin{bmatrix} 1 \\ 2 \\ 0 \\ \end{bmatrix}+w \begin{bmatrix} 1 \\ 3\\ 0\\ \end{bmatrix} $$ So I treated this as $x=...
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32 views

a property of infinite matrices

An infinite matrix $[a_{ij}]_{i,j\in\mathbb{N}}$ is called invertible, if for any convergent sequence $(y_m)$ there exists exactly one sequence $(x_m)$ such that $y_m=\sum_{n\ge 1}a_{mn}x_n$ for all $...
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1answer
27 views

matrix multiplied by rotation matrix on right side and transpose(rotation) on left side

Would a matrix remain un-rotated if it is multiplied by an orthonormal rotation matrix on right side and transpose of same rotation matrix on the left side?
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1answer
40 views

Linear Algebra Eigenvalues and Eigenvectors [closed]

So I have a 2x2 matrix where equation 1(EQN1) is 1 and 2; equation2(EQN2) 2: 4 and 3 The determinant is det(A-λI)=0 When I first solve the eigenvalues I get λ=5, λ=-1 Now this is where I am lost,...
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1answer
41 views

Solve for $x,y,z$ from the linear equations.

The main question is : $$\begin{align} (b+c)(y+z)-ax &= b-c \tag{1} \\ (c+a)(z+x)-by &= c-a \tag{2} \\ (a+b)(x+y)-cz &= a-b \tag{3}\\ \end{align}$$ Solve for $x,y,z$ if $a+b+c\ne0$ ...
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23 views

About the distributive property of matrices

So We all know that matrix operations are distributive, so here is my question.$A^2+AB\\$ and $BA+B^2$ is two matrix operations I have, I know we can do $A(A+B)$ in the first operation but I'm not ...
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2answers
121 views

Matrix decomposition into square positive integer matrices

This is an attempt at an analogy with prime numbers. Let's consider only square matrices with positive integer entries. Which of them are 'prime' and how to decompose such a matrix in general? To ...
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0answers
25 views

exponent of a matrix, equivalent conditions

Let $A=[a_{ij}]$ be a real $n\times n$ matrix. Prove that the following conditions are equivalent: $(1)$ for every $t\ge 0$, all elements of the matrix $\exp (tA)$ are nonnegative $(2)$ $a_{ij}\ge ...
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0answers
57 views

How many subsets of $n$ linearly independent binary strings of length $n$?

Let's consider binary words of length $n$ with elements {-1,1}. There are $2^n$ binary words of length $n$. Now let's consider a subset of $n$ such binary words. All possible subsets are $\binom{2^n}{...
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6 views

May I Know a Algorithm to compute generator polynomial coefficients for RS codes (255,245,t=5) in GF(256)

May I Know a Algorithm to compute generator polynomial coefficients for RS codes (255,245,t=5) in GF(256) ? I want to write a program to compute generator polynomial coefficients for RS codes (255,...
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1answer
40 views

Basis of tensor product of subspaces

Consider two vector spaces $S$ and $S\otimes S$, both of which are subspaces of $H\otimes H$, where $H$ is of $d$ dimension and so $H\otimes H$ is of $d^2$ dimension. We assume that $S$ is of $n$-...
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1answer
24 views

Why are not these two sets subspaces of $\mathbb{R}^3$?

Why are not these two sets subspaces of $\mathbb{R}^3$? $$ \begin{align} S_1&=\left\{\begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}:x_1=x_3\text{ or }x_2=-2x_3 \right\}\\ S_2&=\left\{\begin{...
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1answer
41 views

Solution for an inequality

I want to solve this inequality for $z$ $$(z+1) \left(1-e^x\right)-e^y\geq 0$$ where $-\infty <x\leq \log \left(\frac{1}{z+1}\right)$ and $-\infty <y\leq 0$. I am struggling because $z$ ...
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0answers
42 views

Show that if $d_{i,j} = |i − j|$ then $\sqrt{d}$ is euclidean . [closed]

Is there a standard way to proof that some space is Euclidean? Thanks
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1answer
32 views

Convex basis of functions

I'm looking for a set of convex functions which is forms a basis for $C^1(\mathbb{R})$? Most of the basises I know are polynomials or Fourier basis but I was wondering if there was a basis of convex ...
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1answer
35 views

The set of all real or complex invertible matrices is dense

I'm trying to show that the set of all invertible matrices $\Omega$ is dense over $F=\mathbb R$ or $\mathbb C$. Let $A\in\Omega$ and $C\in M_{n\times n}(F)$. Since $\|A-C\|<\frac{1}{||A^{-1}||}$, ...
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1answer
34 views

Finding $S$, $D$, and $S^{-1}$ such that $A = SDS^{-1}$

Let $A = \begin{bmatrix}18&12\\-40&-26\end{bmatrix}$Find $S$, $D$, and $S^{-1}$ such that $A = SDS^{-1}$ So I did $\det(A-\lambda I)$ to get the char. poly. eqn. and got eigenvalues $\...
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Transpose of matrix (block matrix form) [closed]

Suppose $A$ is a matrix $2N \times 2N$ which is made by a matrix $a, b,c,d$, a $N\times N$ matrix. I want to know following holds \begin{align} &A= \begin{pmatrix} a & b \\ c& d ...
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1answer
25 views

Suppose $A$ is an invertible $n \times n$ matrix and $v$ is an eigenvector of $A$ with associated eigenvalue $4$. Convince yourself that $v$

Suppose $A$ is an invertible $n \times n$ matrix and $v$ is an eigenvector of $A$ with associated eigenvalue $4$. Convince yourself that $v$ is an eigenvector of the following matrices, and find the ...
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3answers
44 views

Is $W=\{A \in M_{n\times n}: \det(A)\neq0\}$ a subspace of $M_{n\times n}(\mathbb{R})$?

How can I prove if two matrices of $W$, say $w_1 ,w_2$, are closed under addition and scalar multiplication. I know that under scalar multiplication $w_1$ is still in $W$ but is there a way to prove $\...
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2answers
24 views

Expressing $v$ as a linear combination of $v_1, v_2, v_3$ and Finding $Av$

Let $v_1 \begin{bmatrix}0\\-2\\2\end{bmatrix}, v_2 = \begin{bmatrix}1\\2\\0\end{bmatrix}$ and $v_3 = \begin{bmatrix}2\\0\\-1\end{bmatrix}$ be eigenvectors of the matrix $A$ which correspond to the ...
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0answers
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Minimal polynomial and possible Jordan forms

Let $A$ be an $8\times 8$ complex matrix with characteristic polynomial $$p_A(x)=(x-1)^4(x+2)^2(x^2+1)$$ and minimal polynomial $$m_A(x)=(x-1)^2(x+2)^2(x^2+1).$$ Determine all possible Jordan ...
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1answer
28 views

Finding formulas for the entries of a matrix

Let $M = \begin{bmatrix}8&2\\-1&5\end{bmatrix}$ Find formulas for the entries of $M^n$ where $n$ is a positive integer $M^n = ?$ (Should be a $2 \times 2$ matrix) What do they mean ...
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1answer
20 views

How can I compute $A(v_1 + v_2)$ where $v_1$ and $v_2$ are eigenvectors of the matrix A

If $v_1 = \begin{bmatrix}5\\3\end{bmatrix}$ and $v_2 = \begin{bmatrix}3\\1\end{bmatrix}$ are eigenvectors of a matrix $A$ corresponding to the eigenvalues $\lambda_1 = -1$ and $\lambda_2 = 4$ ...
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2answers
37 views

Break even methodology

Southeast Moldings molds plastic handles which cost $\$1.00$ per handle to mold. The fixed cost to run the molding machine is $\$3,640$ per week. If the company sells the handles for $\$4.00 $ each, ...
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38 views

finding the solution to $(I_3+A)x=b+2x$

I am having trouble solving the $(I_3+A)x=b+2x$ without finding the matrix $A$. you are also given the inverse of $A$ and the matrix $b$ which consist of 3x1.
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4answers
55 views

Writing the solution set(s) of the equation $Ax = 0$

Consider the following matrix $A = \begin{bmatrix}1&-4&0&0&1\\0&0&1&0&5\\0&0&0&1&1\end{bmatrix}$ a) Write the solution set of the equation $Ax = 0$ ...
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19 views

Geometric interpretation of linear programming dual

Is there a geometric interpretation of the linear programming dual in terms of the primal? I feel like without some sort of intuition of it, I don't truly understand it.
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0answers
32 views

Dot product of two vectors as the eigenvalue of a special matrix [duplicate]

I just noticed that for any two Cartesian vectors their dot product is precisely the only non-zero eigenvalue (if such exists) of the following matrix: $$\vec{a}=(a_1,a_2,a_3,\dots)$$ $$\vec{b}=(b_1,...
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1answer
15 views

Identity relative to different orthonormal bases is unitary

Let $V$ be a finite-dimensional inner product space, and let $\beta,\beta'$ both be orthonormal bases for $V$. Is it the case that $[I]^{\beta'}_{\beta}$ is unitary? If so, how can we prove this? ...
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2answers
31 views

Can a set of four vectors be a basis for P5?

From what I understand, you would need 3 vectors to form a basis of three dimensional space, but does this same restriction apply to a polynomial of let's say P5? In other words, if I'm given W={x^5, ...
0
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2answers
82 views

Tableau and Simplex Method - No Calculator

A non-profit offers crafts complimentary gift packages for its donors. The non-profit costs for each package are \$4 for the Bronze level package, \$7 for the Silver level package, and \$9 for the ...
0
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1answer
34 views

Multiplicative Matrix Functions

What are some examples of multiplicative functions on matrices? More precisely, I'm looking for $f : M^{n \times n}, M^{n \times n} \to R$ with the property $f(AB) = f(A)f(B)$ where A, B are $n \...