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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Calculating the sum $\frac{1}{2} \sum x^T \Sigma x$ for all $x \in \{0,1\}^n$

Note: the equation inside the sum is related to Boltzmann Machines / Hopfield Networks, the energy function of these functions are similar. For further info, i.e. on how to derive the maximum ...
5
votes
1answer
42 views

what can be derived from similar matrix

If $A=\begin{pmatrix} 0&\star&\star \\ \star&x&\star \\ \star & \star & 5 \end{pmatrix}$ is similar to $B=\begin{pmatrix} 1&0&0 \\ 0&y&0 \\ 0 & 0 & 10 ...
1
vote
0answers
96 views

Factoring a polynomial over $\mathbb F_{2^8}$

How do you find the factors of $x^4+x+1$ in $GF(2^8)$ in terms of polynomials? Let me explain, We have primitive irreducible polynomial $p(x)=x^2+x+1$ in $GF(2^2)$ which has root $\alpha^2+\alpha$ in ...
2
votes
1answer
63 views

$n\times n$ matrix with all eigenvalues equal to $1$ or $0$. Does a conjugated matrix with only $1$'s and $0$'s exist?

Let $A$ be an $n\times n$ matrix with all eigenvalues equal to $1$ or $0$. Is there a conjugated matrix $B = XAX^{-1}$ for some $X$ such that all the elements equal either $1$ or $0$? My thoughts so ...
1
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3answers
208 views

How close apart are two message - “Document Distance” algorithm

I was looking at this algorithm that computes how close apart are two texts from one another and the formula seems a bit weird to me. The basic steps are: For each word encountered in a text you ...
0
votes
1answer
49 views

Shifting of the spectrum of a linear operator - in both the symmetric and non-symmetric cases,

a) I finished a problem that sort of highlighted the fact that if a real symmetric matrix $A_2$ = A + I, where A is also real and symmetric, then $A_2$ has the same eigenvectors as A, but its spectrum ...
3
votes
1answer
263 views

Linear programming algorithm that minimizes number of non-zero variables?

I have real world problems I'm trying to programmatically solve in the form of $$Z = c_1 x_1 + c_2 x_2 + \cdots + c_n x_n$$ Subject to \begin{align} & a_{11} x_1 + a_{21} x_2 + \cdots + a_{n1} ...
0
votes
2answers
75 views

Square block matrix, with Hermitian, non-negative definite blocks, prove that the matrix is also non-negative definite,

Consider the square block matrix $$S= \begin{bmatrix} R & RQ^* \\ QR & QRQ^* \\ \end{bmatrix} $$ where $R$ is a Hermitian, non-negative definite square matrix ...
3
votes
1answer
108 views

I can't understand a step in the proof of the associativity of matrix multiplication

Matrix multiplication is proven by the following reasoning: Let there be matrices $A^{m \times n}$, $B^{n \times k}$ and $C^{k \times l}$. Then $$ ...
0
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1answer
83 views

Over-specified linear system

Consider the matrix $A $ with RREF consisting of three of the 4, 4- dimensional standard vectors: $[\mathbb {e_1}, \mathbb {e_2}, \mathbb {e_3} ] $ Since the rank is 3 the matrix has one solution ...
7
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1answer
513 views

Prerequisites for Linear Algebra Done Right by Sheldon Axler.

I've read some notes online and I learned so far: $\{\overset{\displaystyle\ldots}\ldots$ Systems of Two Linear Equations ...
1
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0answers
53 views

matrix with positive diagonal elements

I was wondering if a symmetric matrix with positive elements only in the diagonal (negative elsewhere) is any special beside the symmetry. Thanks in advance
0
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3answers
51 views

Linear independence related with functions

Good day ! I don't understand the following problem: "Prove that the three functions $x^2,\cos{x},e^x$ are linearly independent" So I think so I have to prove that the linear combination: $a ...
1
vote
4answers
168 views

What does this theorem in linear algebra actually mean?

I've just began the study of linear transformations, and I'm still trying to grasp the concepts fully. One theorem in my textbook is as follows: Let $V$ and $W$ be vector spaces over $F$, and ...
2
votes
1answer
121 views

Subset of $\mathbb{R}^4$ such that the intersection with a hyperplane is dense and does not contains $4$ coplanar points.

Does it exist a subset $S$ of $\mathbb{R}^4$ such that for all affine hyperplane $H\subset \mathbb{R}^4$, the set $H \cap S$ is dense in $H$ and does not contains $4$ coplanar points? More than ...
3
votes
1answer
44 views

how to find the dimension of the image of $f$ in this case?

Let $A \in M_{m \times n}(\Bbb R)$ be fixed, and let $B \in M_{m \times l} (\Bbb R)$. Consider the map $f: M_{n \times l}(\Bbb R) \to M_{m \times l}(\Bbb R)$ defined by $f(X) = AX + B$ for all ...
1
vote
1answer
61 views

Determining the standard matrix from the images of the standard basis vectors

Let a linear transformation $T:$ $\mathbb{R}^3$ → $\mathbb{R}^3$ rotate a vector around the z-axis by $45^{o}$ followed by an orthogonal projection onto the x-axis. Determine the standard matrix ...
2
votes
2answers
176 views

Basis for kernel of trace map

Let $T: M_{n \times n}(F) \rightarrow F$ be defined by $T(A) = tr(A)$. I want to find out what a basis is for the kernel $N(T)$ of this linear map. I know $tr(A) = \sum_{i=1}^{n} A_{ii}$. I also ...
2
votes
1answer
96 views

Linear independence question (do 2 vectors who are not multiples of one another and a third which is not in their span form R^3?

The True/False question is: Suppose that $v_1, v_2, v_3$ are in $\Bbb R^5$, $v_2$ is not a multiple of $v_1$, and $v_3$ is not a linear combination of $v_1$ and $v_2$. Then $\{v_1, v_2, v_3\}$ is ...
0
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1answer
65 views

Algebra Problem: Two airplanes flying in opposite directions.

I tried to create the equation and failed. Maybe somebody could help me? Two airplanes start flying from an airport in opposite directions, one averaging a speed of 40 km/hr greater than that of the ...
2
votes
0answers
56 views

Find the solution of linear equation using Wiedemann/ Krylov method

I am using Wiedemann (some literature called Krylov method) to find the solution of a linear equation that defined as $$Mx=b$$ Instead of resolving entire elements of x (size $K \times 1$), we can ...
0
votes
1answer
40 views

Linear ALgebra Subspaces over sets

Let $V$ be a vector space over a field $F$ and $M,N ≤ V$. Consider the following subsets of $V$: (a) $M \cup N$ (b) $M \cap N$ (c) $M+N$ (d) $M - N$ For each of the subsets in (a)-(d) above, ...
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2answers
42 views

Inverse of nonnegative Toeplitz matrice

Consider a right-hand circulant matrice of size $n$ (called also Toeplitz matrice) \begin{equation} T= \left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_n \\ a_n & a_1 & ...
1
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2answers
212 views

Simultaneously diagonalization of two matrices.

Let $A$ be a real symmetric matrix and $B$ a real positive-definite matrix. Is it possible to simultaneously diagonalize of $A$ and $B$? Thank you very much.
0
votes
1answer
27 views

expressing canonical base of univariate polynomials in binomial base

Two bases are fairly standard for ${\mathbb Q}[X]$ : the canonical base $(X^j)_{j\geq 0}$ and the binomial base $(b_j(X))_{j\geq 0}$ where $b_j(X)=\binom{X}{j}=\frac{X(X-1)\ldots (X-(j-1))}{j!}$ (thus ...
6
votes
1answer
168 views

Eigenvalues of symmetric matrix with skew-symmetric matrix perturbation

If $A$ is diagonalizable, using the Bauer-Fike theorem, for any eigenvalue $λ$ of $A$, there exists an eigenvalue $μ$ of $A+E$ such that $|\lambda-\mu|\leq\|E\|_2$ (the vector induced norm). Here I ...
0
votes
4answers
95 views

$A.A^t$ is diagonal

Be $A$ a semidefinite nonnegative matrix. What kind of conclusions can we say about $A$ if $A.A^t$ is diagonal? Same question when $A$ is binary matrix. Thanks
2
votes
1answer
73 views

Künneth formula in topology, show isomorphism

Where could I find a proof of the isomorphism aspect of Theorem 2.4 in this pdf: http://math.stanford.edu/~conrad/diffgeomPage/handouts/tensor.pdf For vector spaces $V$ and $W$, consider $V$ and ...
2
votes
1answer
83 views

How to show trace of $AB$ is zero for $A \in \mathfrak{u}_n$ and $B \in \mathcal{H}_n$?

Please have a look at this question: Help needed in understanding the basics of Cartan decomposition of a Lie algebra I want to show that the decomposition $\mathfrak{gl}_n = u_n \oplus ...
8
votes
1answer
120 views

If $A$ and $B$ are $n×n$ matrices such that $AB=B$ and $BA=A$ then find the value of $A^{4} + B^{4} - A^{2} -B^ {2} + I$

The given question is If $A$ and $B$ are $n×n$ matrices such that $AB=B$ and $BA=A$, then find the value of $A^{4} + B^{4} - A^{2} -B^ {2} + I$. Any hints?
1
vote
1answer
72 views

If $A$ is Hermitian and $B$ is skew-hermitian matrix, then trace of $AB$ is zero?

I reduced a problem to showing that if $A \in End (\mathbb{C}^n)$ is such that $A=-\bar{A}^t$ and if $B\in End (\mathbb{C}^n)$ such that $B=\bar{B}^t$ then the trace $Tr(AB)=0$. But I cannot show ...
6
votes
1answer
2k views

Is Hoffman-Kunze a good book to read next?

I'm planning on self-studying linear algebra, and trying to decide on a book. I'm thinking of using Hoffman and Kunze. What sort of experience is required to handle Hoffman and Kunze? So far, I've ...
0
votes
1answer
67 views

Finding the eigenvalues and eigenvectors of $A^{n}$

Find the eigenvalues and eigenvectors of $A^{5}$ for $A = \begin{bmatrix} 0&0&-1 \\-1&1&-1 \\ 1&-1&0\end{bmatrix}$. How many eigenspaces does it have? What is the dimension ...
2
votes
1answer
77 views

show $\int_0^1 f(t)g(t) dt$ is a non-degenerate scalar product

Let $C[0,1]$ be the vector space of continuous real-valued functions on the interval [0,1]. The following mapping is defined: $$ \langle \bullet , \bullet \rangle : C[0,1]^2 \rightarrow \mathbb{R} : ...
1
vote
1answer
77 views

Self-adjoint matrices: prove that $\operatorname{Tr}\left((AB)^2\right)\le\operatorname{Tr}\left(A^2B^2\right)$ [closed]

$A,B \in M_n(\mathbb C)$ and self-adjoint. Prove the following inequality: $\operatorname{Tr}\left((AB)^2\right)\le\operatorname{Tr}\left(A^2B^2\right)$. Thanks
0
votes
2answers
46 views

Another Look at Systems of Equations: Possible Solutions

This is more of a general question than a specific problem. While doing homework, I am being asked to find the solutions for complicated systems of equations, and I was wondering: Suppose I were to ...
2
votes
2answers
72 views

linear map $f:V\rightarrow V^*$ or $\mathbb F$.

I'm having a bit of trouble understanding the dual space $V^*$ to a vector space $V$ over field $\mathbb F$. So far I understand that a linear form/functional $f$ is a linear map from $V$ to its ...
0
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2answers
59 views

Finding all matrices for which the homogeneous system has a given solution space

Find all $3\times 3$ matrices for which the homogeneous system has a solution space as the line $x = 2t$, $y = t$, $z = 0$. (Hint: Write the row reduced augmented matrix from given information.) ...
3
votes
2answers
76 views

Given $A$ is $6×6 $ real symetric matrix of rank $5$ , then to determine rank of $A^{2}+ A+I $

Given $A$ is $6×6 $matrix of rank $5$ , then to determine rank of $A^{2}+ A+I $. I knowthat rank of matrix doesnot change when we square it , but how to proceed in this question.Any hints ? Thanks
1
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1answer
82 views

Basis if and only if $\varphi$ is an isomorphism

Let $V$ be a finite dimensional vector space. I am aware of the theorem stating that if $\varphi \colon V \rightarrow V$ is an automorphism and $\mathcal{A}=(a_1, \ldots, a_n)$ is a basis of V then ...
6
votes
4answers
506 views

Proof If $AB-I$ Invertible then $BA-I$ invertible.

I have these problems : Proof If $AB-I$ invertible then $BA-I$ invertible. Proof If $I-AB$ invertible then $I-BA$ invertible. I think I solve it correctly, But I'm not so sure, I'll be glad to ...
1
vote
1answer
62 views

What do we know about the solution of this set of linear equations?

Let $C \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times n}$, $Y \in \mathcal{Y}$, and $B : \mathcal{Y} \to \mathbb{R}^n$ be linear, where the linear space $\mathcal{Y} \subset \mathbb{R}^m$ may be ...
4
votes
2answers
102 views

Cube roots escape [closed]

$ \sqrt{\sqrt[3]{5}-\sqrt[3]{4}} \times 3 = \sqrt[3]{a} + \sqrt[3]{b} - \sqrt[3]{c}, $ where $ a, b $ and $ c $ are positive integers. What is the value of $ a+b+c $? This question appeared in one ...
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0answers
85 views

Request a paper by Gelfand and Ponomarev

I am looking for the following paper by Gelfand & Ponomarev: I. M. Gelfand and V. A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a ...
3
votes
1answer
32 views

What this vector equation means

I have this equation K = (A, B, C) / |A x B| A, B, C are 3D vectors. K should be number, not vector. How is this calculated, or what this expression means?
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1answer
49 views

Basis for the space of quadratic polynomials orthogonal to those with $p(2)=p(1)$

Let $P_2[x]$ be the space of polynomials of degree less than or equal to 2. If $W = \{p ∈ P_2[x] \mid p(2) = p(1)\}$, then find a basis for $ W^⊥$ where $P_2[x]$ is equipped with an inner product ...
0
votes
1answer
186 views

How do row operations affect the column space?

I've been curious about this: Row operations do not affect the row space, but they affect the column space. Is there any way to 'systematically' perform row operations to make the column space the ...
4
votes
1answer
82 views

Matrix equation solution

Does anybody know how to solve this matrix equation: $$P = P P^T R + X,$$ where $P, R,$ and $X$ are vectors with $n$ elements, and $P$ is the unknown vector?
0
votes
1answer
356 views

Quadratic forms, diagonal form, and whether an orthogonal transformation exists for a matrix,

a) Let $$ \begin{bmatrix} 3 & 2 & -2 \\ 2 & 3 & -2 \\ -2 & -2 & 5 \\ \end{bmatrix} $$ be a quadratic form. Write explicitly an ...
1
vote
2answers
44 views

Showing $(Tp)(x) = x^2p(x)$ is a linear map (transformation)

Define a linear map function $T: \mathcal{P}(\mathbb{R}) \to \mathcal{P}(\mathbb{R})$ where $\mathcal{P}(\mathbb{R})$ is the set of all polynomials with real-valued coefficients. Now let $T$ belong to ...