Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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interger matrices whose mth power is identity

how can one find all the matrices with integer entries of size $n \times n$ such that $A^{m}=I$ where $m$ is fixed integer and the matrix does not have fixed point in $\mathbb{Z}^n$ (except zero of ...
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1answer
29 views

Abelian group over a field underlying an abstract vector space

Given that a set V is said to be a vector space over a field F if V is an Abelian group under addition and for each $a\in F$ and $\boldsymbol{v}$ in V there is an element $a\boldsymbol{v}$ in V, how ...
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3answers
33 views

Find a basis of $A = (\{1, \sin(x), (\cos)^2(x), (\sin)^2(x)1\})$ and determining its dimension.

We consider a space F(R,R) of functions of R in R. Let A = ({1, \sin(x), $\cos^2(x)$, $\sin^2(x)$}) Find a basis of the vector subspace of F(R,R) and determine its dimension. So I used the identity ...
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1answer
17 views

Ring structure on tensor product of two $A$-modules

Let $A, B, C$ be a commutative rings. Suppose I have two ring homomorphisms, $\alpha : A \rightarrow B$ and $\beta : A \rightarrow C$. I am trying to show that $B \otimes_A C$ has a ring structure ...
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14 views

Rank of product of two matrices [duplicate]

I want to show that $\text {rank} ( AB)\le \min(\text{rank} B, \text{rank} A) $ and when the equality occurs? Please help me with this problem by giving hints, solving, or suggesting a book. Thanks
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24 views

A coherence question in rigid monoidal categories

Let $\mathscr{C}$ a monoidal symmetrical closed category and $t_{A, B}: [A, B] \otimes B \to A$, $u_{X, Y}: X \to [Y, X \otimes Y]$ the co-unit and unit of the adjunction $(A \otimes B, C) ...
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3answers
31 views

What is the cardinal of a field F_5 vector space of dimension 3?

What is the cardinal of a field F_5 vector space of dimension 3? The mark scheme says since F_5 = { 0,1,2,3,4 } there are 5 possibilities. so it is 5^3. So the card(v) = 125. But in the lecture ...
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1answer
44 views

$3 \times 3$ real matrix: relation with determinants

$A$ is a $3 \times 3$ matrix with real entries such that $\operatorname{det}(A+I_3)=\operatorname{det}(A+2I_3)$. Then is $2\operatorname{det}(A+I_3)+\operatorname{det}(A-I_3)+ 6 =3 ...
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2answers
239 views

Problems on Symmetric Matrices

1 . Let $A = (a_{ij})$ be a real $n \times n$ matrix such that $a_{ij} = a_{ji}$ for all $1 \leq i,j \leq n$ and $a_{ij} = 0$ for $|i-j|>1$. Moreover $a_{ij}$ is non-zero for all $i$,$j$ satisfying ...
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1answer
18 views

permutations with a given condition!

What will be the number of permutations of n different things, taken r at a time,when p particular things is to be always included in each arrangement? I know the answer to this question but could not ...
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1answer
19 views

Does an isotropic vector always exist for an indefinite quadratic forms?

I have some problem reading some paper. In that paper, the author proved that a quadratic form $Q$ has an isotropic vector(of course, nonzero) by showing $Q$ has both nonnegative and nonpositive ...
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30 views

Minimum and maximum determinant of a sudoku-matrix

Let A be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of A ? A must have the dominant eigenvalue 45, but this ...
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1answer
69 views

Notation for dimension of vector space

Is it an unusual notation to write $|V|$ for the dimension of a vector space $V$? Is it ok to use it if you blur the distinction between the grid for the finite element method and its associated ...
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0answers
13 views

Matrix multiplier for ODE

I have matrix C with dimensions $3 \times 3 $ and it is skew symmetric too C is given by $C(0,0)=0,C(1,1)=0,C(2,2)=0 \tag 1$ $C(1,0)= sc_0+ px (c_1-c_0),C(0,1)=-C(1,0) \tag 2 $ $C(0,2)= ...
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0answers
23 views

Solving tridiagonal matrices where the top left element is zero

If I have a matrix like this: $$ \left[\begin{array}{rrrrrrrrr|r} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & ...
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1answer
33 views

An isomorphism question

Suppose $V$ is a vector space and $W$ is a linear subspace of $V$,can we conclude that $$W\oplus (V/W)\cong V$$ It looks simple but I can't see any formal proof... Any hints or book ...
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2answers
392 views

Ways Of Matrix Multiplication

Let $A \in F^{11 \times10}$ and $B\in$ $F^{10\times11}$ We only know $2$ rows of $A$ and $3$ columns of $B$. How many entries of $B\cdot A$ can we know? I think the answer is none, because there are ...
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25 views

Orthogonal Transformation (rotation to new axes)

I need to find the linear transformation that maps the points the standard cartesian coordinates to ones where the z axis is in the direction of $(1, 1, 1 )$. It needs to also preserve the ...
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1answer
26 views

Derivative of logistic loss function

I am using logistic in classification task. The task equivalents with find $\omega, b$ to minimize loss function: That means we will take derivative of L with respect to $\omega$ and $b$ (assume y ...
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2answers
279 views

Symmetric matrix multiplication

Let $A$ and $B$ be symmetric matrices. Prove: $AB=BA$ $AB$ is a symmetric matrix As for 1. due to the axiom $(AB)^T=B^T A^T$ so $AB=BA$ As for 2. I did not find any axiom that can support the ...
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1answer
21 views

Express summation in terms of matrix norm

Express the following $$\sum _{ i=1 }^{ n }{ ({ \beta }_{ 1 }x_{ i }+{ \beta }_{ 0 }-y_{ i })^{ 2 } }$$ To become something of the form: $∥Ax−b∥^{ 2 }$ where $A$ is an $m$−by−$n$ matrix and $b$ is ...
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1answer
21 views

Matrix transpose times itself

We define A to be a matrix in $R^{m*n}$ Does $A^TA$ have any particular structure? When is $A^TA$ invertible?
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1answer
27 views

Find the Jacobian of F

Given that $A \in \mathbb{R}^{m\times n}$, and $b \in \mathbb{R}^{m}$, we define: $$F:\mathbb{R}^{n} \rightarrow \mathbb{R} = \left\| Ax-b \right\|^2$$ Find the Jacobian of $F$, and show that it is of ...
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Bayesian linear regression cost function

I am studying classification using linear regression . Now, I want to map it in Bayesian regression. Let talk about binary classification using linear regression again. Assume that I have a set ...
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3answers
45 views

Suppose that $V_1$ and $V_2$ are subsets of a vector space…

Suppose that $V_1$ and $V_2$ are subsets of a vector space, is $span(V_1\cup V_2) = span(V_1)\cup span(V_2)$? This seems like it should be pretty straight-forward but something is baking my noodle. ...
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42 views

How prove $S_{k}(x)=\sum_{i=1}^{n}x^k_{i}$ this System of equations The only solution?

when I read a china book,I see this follow interesting problem (the author says it is clear have follow) if give the number $S_{k}(x),k=1,2,3,\cdots,n$ ,and such $$\begin{cases} ...
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1answer
20 views

Relation between condition numbers $\kappa(A^T A)$ and $\kappa(A)$

Let $A$ be a real $m\times n$ matrix. Why is the condition number $\kappa(A^T A)$ approximately the square of the $\kappa(A)$?
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41 views

Copy of C in H , trace is independent of the choice

Let X€ Mn(H). For each of the choices of a copy of C in H , write out the corresponding matrix of X as an element of M(2n,C). Use this formula to show that the trace of X is independent of the choice. ...
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2answers
100 views

Show that exponential map is surjective

How I can show that $\exp\colon \mathcal M(n,\mathbb C) \rightarrow \text{Gl}(n,\mathbb C)$ is surjective? Thank you.
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4answers
52 views

Determine whether the set $W$ is a subspace in $\mathbb{R}^{3}$

Determine whether the set $W=\{(2a-2,3b,2a-3b)\}$ is a subspace in $\mathbb{R}^{3}$. Describe the set. I have tried putting the set into matrix form but don't know which is correct. I also know that ...
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33 views

Linear Algebra subspaces [duplicate]

Determine whether the set $W= \{(2a-2,3b,2a-3b)\}$ is a subspace in $\mathbb R^3$ Describe the set. I know that in order for a set to be a subspace, it must be closed under multiplication and ...
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1answer
11 views

Determine if matrix D belongs to Vect(A,B,C)

So there are 4 matrices, A, B,C,D. They belong to field F5. Determine if D belongs to Vect(A,B,C). I have pretty much done all the calculations its just i fail to conclude/find the right value for the ...
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17answers
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If $AB = I$ then $BA = I$

If $A$ and $B$ are square matrices such that $AB = I$ where $I$ is identity matrix. Show that $BA = I$. I do not understand anything more than the following. Elementary row operations. Linear ...
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0answers
16 views

Matrix norm to compare two graphs

I have the adjacency matrices of two undirected graphs. I want to measure how different the two matrices are in terms of the linkage. Both matrices have the same number of nodes, but they differ in ...
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1answer
26 views

LDU matrix decomposition

Let $A$ be a matrix that can be written as $LDU$ for some lower unitriangular matrix $L$, some diagonal matrix $D$ and some upper unitriangular matrix $U$. Then, are the eigenvalues of $A$ the same as ...
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change some element of a correlation matrix

I am working on correlation matrices. These matrices have the main property to be symetric , positive-semidefinite, have 1 on the diagonal and each of their elements is between -1 and 1. Let's say I ...
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1answer
107 views

Real inner product from a complex one

Let $V$ be a complex vector space. We may view $V $ as a real vector space by simply ignoring non-real scalars. Now suppose that $\langle \cdot,\cdot,\rangle$ is a complex inner product on $V$, and ...
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1answer
49 views

Fast way to inverse B'CB+D

$\mathbf {A = B'CB}$, where $\mathbf A$ is of dimension $n \times n$, $\mathbf C$ is m by m, positive definite and symmetric, $\mathbf B$ is of dimension $m \times n$, and $n >> m$. Inversion ...
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1answer
22 views

Reduced Row Echelon form without scalar multiplication?

Is it possible to transform any matrix to row reduced echelon form without using the row operation that multiplies a row by a scalar?
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1answer
38 views

Do $Ax=b$ When $A=I$ Must Always Have One Solution [on hold]

Let there be $Ix=b$. So every unknown have the same or different value there for every $Ix=b$ there will be always one answer solution?
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47 views

Zero set of finitely many polynomials.

Somebody asked a question earlier regarding this proof but I'm confused about a different part. I understand everything but the line "As the zero set of finitely many polynomials, $R$ is a closed ...
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0answers
17 views

Usual linear combination and the one with measure

Let $X$ be a Borel measurable subset of $\Bbb R^n$ and let $\nu$ be a probability measure on $X$. Can we always find an integer $m$, points $x_1,\dots,x_m\in X$ and coefficients $a_1,\dots,a_m \geq ...
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1answer
28 views

Matrix Semi-Definite Inequality

Does the following inequality hold? If matrix $A$ is a $n \times n $ positive semi-definite, $A \succeq 0$, and $U$ is one $n \times k$ unit column-orthogonal matrix ($k \leq n$), $U^{T}U=I$, do we ...
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19 views

Interesting properties of functions and sets that depends on dimension of space.

For $n=1$ (or $m=1$), we have some basic properties of functions and sets that are not valid (or not necessarily valid) for $n\neq 1$ (or $m\neq 1$). For exemple: Calculus. Let $[a,b]$ be a closed ...
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1answer
57 views

What is the name of the matrix that is created by a vector times its transpose.

I am looking for the name of the matrix created by the following operation: $Z = z*z^T$ I know it should create a symmetric matrix with an element $Z_{ij} = z_{i}z_{j}$
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Top bound on the value of an algebraic adjunct to elements of a nonnegative irreducible matrix

Let $A = ||a_{i j}||_1^n$ be nonnegative irreducible matrix with maximum eigenvalue $r$. Let $A_{i j}(\lambda)$ be an algebraic adjunct for the element $\lambda \delta_{i j} - a_{i j}$ in determinant ...
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1answer
23 views

logarithm of projection

I want to prove what's used in the fourth line below the "Proof" section here: http://en.wikipedia.org/wiki/Quantum_relative_entropy#The_result The statement is: Let $\rho$ be a density operator on a ...
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0answers
17 views

finding the symmetric point

let there be $4$ points. $A(-1,1,1), B(2,0,-1), C(1,3,-2), D(-2,-1,0)$. the $4$ points are not on the same line. the plane which goes through the points $A$ and $B$, and which is also paralel to the ...
3
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7answers
168 views

If $A^2 = B^2$, then $A=B$ or $A=-B$

Let $A_{n\times n},B_{n\times n}$ be square matrices with $n \geq 2$. If $A^2 = B^2$, then $A=B$ or $A=-B$. This is wrong but I don't see why. Do you have any counterexample?
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1answer
981 views

High-level linear algebra book

Please, recommend high-level and modern books on linear algebra (not for first reading). Like Kostrikin, Manin "Linear algebra and geometry" or respective chapters of Lang "Algebra".