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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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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11 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
19 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
29 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
151 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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17 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
21 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
109 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
18 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
9 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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1answer
26 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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0answers
13 views

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
44 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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0answers
40 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} ...
2
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1answer
18 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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40 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
98 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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2answers
51 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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2answers
32 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
9 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 ...
67
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17answers
13k views

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
12 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
23 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
47 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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0answers
46 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
16 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
27 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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0answers
18 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 ...
4
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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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0answers
23 views

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
21 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
16 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 ...
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0answers
22 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 ...
3
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7answers
164 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".
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1answer
31 views

Maps preserving roots of a polynomial function over finite fields

Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$. Let $S(P)=\{ ...
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1answer
34 views

How to show that $w$ is a $N$th primitive root of unity?

I am studying the discrete Fourier transform. For sequence $x_{0}, \dots, x_{N-1}$ it is defined as $$X_{k} = \sum_{n=0}^{N-1} x_{n}e^{-2\pi ikn/N} \quad 0 \leq k \leq N-1$$ according to Wikipedia. ...
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15 views

Prove there exists T, supposing that T does not hve a diagonal matrix with respect to any basis of C3.

Suppose $T$ in $L(\mathbb C^3)$ is such that $6$ and $7$ are eigenvalues of $T$. Furthermore, suppose $T$ does not have a diagonal matrix with respect to any basis of $\mathbb C^3$. Prove that there ...
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1answer
21 views

Eigenvalues, Eigenvectors, and Invariant Subspaces [on hold]

Suppose $T \in L(F^5)$ and $\mathrm{dim}~ E (8,T)=4$. Prove that $T-2I$ or $T-6I$ is invertible.
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2answers
56 views

Determinant of an ill conditioned matrix

I have the following ill conditioned matrix. I want to find its determinant. How is it possible to calculate it without much error \begin{equation} \left[\begin{array}{cccccc} ...
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2answers
30 views

Eigenspace, Diagonalizable, Direct Sum

Suppose $T$ is an element of $L(V)$. Prove that if $T$ is a diagonalizable operator, then $\mathrm{null}(T)$ and $\mathrm{range}(T)$ are direct sum of $V$.
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1answer
14 views

Sequences length for LFSR when polynomial is reducible

An LFSR with polynomial 1+x4+x5 = (1+x+x2)(1+x+x3) can generate several sequences, depending on the initial value. If I did not made any mistake enumerating them, the sequences length are 3, 7 and 21. ...
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1answer
35 views

Extension to the complex numbers for ex. 12 in ch. 6 of Axler's “Linear Algebra Done Right”

I'm wondering how the answer to Sheldon Axler's exercise 12 of chapter 6 "Linear Algebra Done Right" changes when the underlying field is extended from the reals to the complex numbers. The exercise ...
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24 views

Eigenvalue bounds for a positive semidefinite matrix

I have a symmetric $(p\times p)$, positive semi definite matrix $\Omega$. If somebody says: find the eigenvalue bounds of the matrix such that $$w_1I \le \Omega \le w_2I$$ where $I$ is the identity ...
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1answer
37 views

Orthogonal Operator Infinite Dimensional Inner Product Space

I know that on a finite dimensional inner product space, a unitary (or orthogonal) operator preserves the inner product. That is, having $\|T(x)\|=\|x\|$ for all $x\in V$ is equivalent to having ...