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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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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1answer
33 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
28 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
42 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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0answers
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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1answer
30 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 ...
4
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0answers
58 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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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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0answers
18 views

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
18 views

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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2answers
30 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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1answer
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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0answers
20 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, ...
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2answers
84 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 ...
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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 \...
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1answer
37 views

confusion in a linear algebra theorem

Insights about $Tv_j=w_j$, the linear maps and basis of domain. I have a question about the theorem mentioned in the link above. I understand what the theorem is saying, but a little uncertain. it ...
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0answers
20 views

Pseudo-inverse of matrix representation is matrix representation of pseudoinverse

Let $T: V \rightarrow W$ be a linear map on finite dimensional inner product spaces $V,W$. Let $\beta,\gamma$ be ordered (orthonormal?) bases for $V,W$ respectively. Is it necessarily the case that ...
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1answer
44 views

Proving/verifying dimension and basis

I'm coming from a computer science background and am currently trying to formalize my linear algebra knowledge by going through Linear Algebra Done Right. I have an intuitive grasp on most of the ...
2
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1answer
34 views

Derivation of gradient for non negative matrix factorization

I am looking at a paper for non-negative matrix factorization and can't seem to figure out the derivation for the gradient. The function is as follows: $f(W,H) = \frac{1}{2}||V-WH ||^2_F$ Where V ...
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2answers
44 views

Which of the following are subspaces of $\mathbb{R}^3$?

I have two examples directly from my book: $$\{(x, y,z) : x + y + z = 1 \}$$ and $$\{(x, y, z) : x \leq y \leq z\}$$ The book once again isn't helping me understand the concept. What are the ...
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2answers
39 views

Is it correct this reasoning?

Let $E,F$ be reals vector space. Since (1) $\dim (E\times F)=\dim E + \dim F$ (2) $\dim\ \text{Hom}(E,F)=\dim E\cdot \dim F$ Given $r>0$ integer, is it true that: $$\text{Hom}(E\times \stackrel{(...
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$Hom(E\times\stackrel{(r)}{\ldots}\times E,E)$ isomorphic to $\bigotimes_r^1 E$?

Let $E$ be a $n$-dimensional $\mathbb{R}$-vector space. Prove that: $$\begin{array}{ccll} \Phi:&Hom(E\times \stackrel{(r)}{\ldots} \times E,E)&\longrightarrow&\bigotimes_r^1 E\\ &\psi &...
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2answers
50 views

I am confused by the statement “the null space of A is a nontrivial”

Correct me if I'm wrong but if a null space of a matrix A is nontrivial would it be correct to say that it is the opposite of the list of points in the Invertible Matrix Theorem? A is an invertible ...
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1answer
49 views

Why all vector space have a span set?

I thought about this question, but I don't sure if my proof is correct. In the book, he put this question like a observation of span sets' definition, so I tried proof this. My attempt: Suppose that ...
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1answer
23 views

A basis for a tensor product space where the tensor elements are linearly dependent

Say I have a space $V^{(1)}$ with basis $\{a_i \}$ and $V^{(2)}$ (with dimensions $d_1$, $d_2$ respectively) with basis $\{b_j\}$. Clearly the vectors $\{a_i\otimes b_j\}$ are a basis for $V^{(1)}\...
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1answer
46 views

How to solve out *SPECIFIC* variables from a linear system?

I want to reduce the size of a linear system by solving out some columns. For example: $$ \mathbf{Ax = b} \\ \text{where } \mathbf{A=} \left[ \begin{array}{cccc} 5 & 1 & 0 & 0 \\ 0 &...
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3answers
81 views

$\ker ST=\ker T$

Let $S$ and $T$ be linear maps between vector spaces such that the composition $ST$ makes sense. Clearly, $\ker ST\supseteq \ker T$. The two instances that come to my mind for having an equality in ...
4
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2answers
74 views

The relation between axes of 3D rotations

Let's suppose we have two rotations about two different axes represented by vectors $v_1$ and $v_2$: $R_1(v_1, \theta_1)$, $R_2(v_2,\theta_2)$. It's relatively easy to prove that composition of ...
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0answers
35 views

Problem of Apollonius with 3 circles of equal radius

I want to find the circle which exclusively touches 3 other circles. This is essentially the classic problem of Apollonius. I use the following equation to find the center of that circle and its ...
2
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0answers
64 views

Is there a concept of “Cross determinant”?

Suppose $A = \begin{bmatrix}a & b \\ c & d \\\end{bmatrix}$. The determinant of $A$ is $$\det A = ad - bc.$$ Suppose $B = \begin{bmatrix}e & f \\ g & h \\\end{bmatrix}$. Now one could ...
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51 views

A nonnegative vector orthogonal to tensor space

Let $S$ be a subspace over $\mathbb{R}$. For a vector $v$, $v\geq 0$ means each entry of $v$ is nonnegative. Does it hold that, for any $v\geq 0$ such that $ v\bot S^{\otimes n} $, there is some $v'\...