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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Generate a random neutrally stable matrix

I need to generate random real matrices such that all eigenvalues have real part equal to 0 -- i.e. random neutrally stable matrices. What's the simplest way to do this? Note that I don't care about ...
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461 views

On Learning Tensor Calculus

I am highly intrigued in knowing what tensors are, but I don't really know where to start with respect to initiative and looking for an appropriate textbook. I have taken differential equations, ...
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2answers
89 views

Prove that the set of bases is linearly independent

Suppose that $W$ and $W'$ are subspaces of the vector space $V$ with the property that $W\cap W'=\{0\}$, and suppose that $\beta$ is a basis for $W$ and $\beta'$ is a basis for $W'$. Prove that the ...
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3answers
263 views

A vectorspace over an infinite field is not a finite union of proper subspaces?

Show that if V is a vector space over an infinite field F, then V cannot be written as set-theoretic union of a finite number of proper subspaces.
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252 views

Give an example of a matrix A such that im(A) is the plane with the normal vector [1,3,2] in $R^3$

Give an example of a matrix A such that im(A) is the plane with the normal vector \begin{bmatrix}1\\3\\2\end{bmatrix} in $R^3$ How would I go about doing this question. The solution manual doesn't ...
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16 views

trying to solve a systems of equations with one inequality

I am trying to create a website that would run off this mathematical formula. I have tried to solve it but I got that there was no answer. I am only in pre-algebra and want a second opinion on if I ...
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2answers
43 views

Simultaneous equations with three parts

\begin{align*} 6a +24b +18c &= 168\\ 8a +28b +22c &= 208\\ 4a + 20b +20c &= 140 \end{align*} I've tried doing this multiplying so they cancelled out but I've always gotten decimal point ...
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36 views

What is the dimension of $f(A)$ if $f:A\subseteq\mathbb{R}^k\to\mathbb{R}^n$ is linear and $A$ is a subspace of $\mathbb{R}^k$?

Let $k\le n$ and $f:A\subseteq\mathbb{R}^k\to\mathbb{R}^n$. Obviously, $f(A)$ is a subspace of $\mathbb{R}^n$ iff $\forall x,y\in A:\exists z\in A:f(z)=f(x)+f(y)$ $\forall x\in ...
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2answers
228 views

derivative of matrix function with kronecker product

In the derivation of an estimator, I'm meant to find the minimum of the following matrix scalar function: $\underset\beta {argmin}$ $[S Y^\prime M^\prime - SX^\prime (kron(I_N,\beta) ) M^\prime ...
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57 views

Vector Space And Sub Space

A vector space need to follow the 8 axioms, while sub-space needs to be non-empty and closed to linear combinations. So for a given vector set, all I can do is to prove that it is a sub-group of a ...
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40 views

A question about adjoint matrices

Let $T:V \to V $ be a linear map on complex vector space $V$ which is equipped with complex inner product $ <. , .> $ we know there exists a unique linear operator $T^* : V \to V $ such that ...
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32 views

Order of composition when dealing with transformations

I have been struggling with a question in my book. $T$ is a translation of $(+5,+4)$, $M$ is a reflection in the line $y=x$. $R$ is a 90 degree anticlockwise rotation about $(0,0)$ Write down ...
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2answers
57 views

Uncountable Subset of Reals Generates Reals by Finite Integral Linear Combinations

A question that I thought of earlier today that I couldn't quite get anywhere with. Given an uncountable subset of the reals, $S$, is it always possible for any $r \in \mathbb{R}$ that we can take a ...
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0answers
58 views

cohomology of general linear group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $\mathrm{GL}_n(\mathbb{Z}_2)$ be the group consisting of all $n\times n$ matrices with entries in $\mathbb{Z}_2$ with non-zero determinant. What is the ...
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1answer
51 views

Correspondence between wedge product and its dual

Consider the map $$ \bigwedge\nolimits^{\!k}(V^*)\to \left( \bigwedge\nolimits^{\!k}V \right)^*\\ \left( \sum_{i=1}^n (f_{1}^i \wedge \dotsb \wedge f_k^i) \right) \mapsto \left( v_1 \wedge \dotsb ...
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1answer
138 views

Wedge product of maps

If $V$ and $W$ are $\mathbb{F}$ vector spaces, A $k$-multilinear alternating map $V^k \to W$ induces a unique linear map $f: \bigwedge^k V \to W$. In the special case $W = \mathbb{F}$ and $\dim V ...
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1answer
25 views

doubt regarding basis of vector space

i am studying basis of a vector space .let vector space be $V$ and subset be $B$. in it the two condition stated were that $B$ is a maximal linearly independent set in $V$ and second condition was ...
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1answer
30 views

Show that all the entries of $x$ are rational numbers

Let $A$ be a non zero $n×n$ invertible matrix with integers entries. Let $b$ a non zero $n×1$ vector with integers entries. Let $x$ be a $n×1$ vectors verifying: $$Ax=b$$ My question is: Show that ...
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191 views

Given an irreducible representation, is there a *unique* unitary representation that it is equivalent to?

I might need help here in understanding my own question in places and please don't hesitate in asking for edits and clarifications. Background: A representation $\rho$ of a finite group $G$ is a ...
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1answer
191 views

Reference: Continuity of Eigenvectors

I am looking for an appropriate reference for the following fact. For each $X \in \mathbb{R}^{n \times n}_{\text{sym}}$ (symmetric matrix), there exist $\varepsilon, L > 0$, such that for ...
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1answer
36 views

Inner Product over real space and complex space

Let $V$ be $n$-dimensional $\mathbb{R}$-vector space with inner product $\langle \,\,,\,\,\rangle_1$. Let $W=V\oplus V$. Then $W$ can be made into a $\mathbb{C}$-vector space by defining addition as ...
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1answer
62 views

Eigenvalue of Orthogonal Transformation

I am solving the question If $\lambda$ is (complex) eigenvalue of an orthogonal transformation $T\colon \mathbb{R}^n\rightarrow \mathbb{R}^n$ then $|\lambda|=1$. Here $\lambda$ is complex eigen ...
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1answer
383 views

Do entries in augmented columns count as pivot?

I am in a basic linear algebra course, and we are learning to solve linear equations with augmented matrices. We learned that when an augmented matrix is in row echelon form or reduced echelon form, ...
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1answer
24 views

Linear automorphisms with a single eigenvector

Let $V$ be a finite dimensional vector space and $GL(V)$ be the automorphism group of $V$. I need to find an automorphism of $V$ with only a single eigenvector for a bit of a proof I'm working on, but ...
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22 views

An ordered basis can be viewed as a linear isomorphism

Let $E$ be a vector space. What is the meaning of the following sentence: An ordered basis on $E$ can be viewed as a linear isomorphism $$p:\mathbb R^k\to E$$
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50 views

How i compute the determinant of a matrix?

Let $A$ be a $3\times 3$ matrix such that $A^{-1}=I-2A$. I want to compute the determinant $A$. I can reply to the question if $A$ be a $2\times 2$ matrix. Since from $A^{-1}=I-2A$ deduce that ...
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0answers
52 views

Orthogonal projectors on non-orthogonal subspaces

It is a well known fact that if(f) $V,W$ are orthogonal subspaces of a Hilbert space $H$, then their orthogonal projectors satisfy: $$ P_{VW} = P_V + P_W, $$ where $P_{VW}$ is the projector on $V+W$. ...
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1answer
113 views

Linear map invertible if and only if associated matrix invertible

Theorem: Let $V$ and $W$ be finite dimensional vectorspaces with ordered bases $\beta$ and $\gamma$ resp., and let $T: V \rightarrow W$ be linear. Then $T$ is invertible if and only if the associated ...
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2answers
107 views

Least squares minimization of point distances (nonlinear)

We have two sets of 2D points $\bar{x}\leftrightarrow \bar{x}'$ (the bar denotes a vector, i.e. $\bar{x}=(x,y)^{T}$). I would like to minimize discrepancy between the points using the least squares ...
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4answers
130 views

Transpose Operator is diagonalizable?

Let $T \colon \mathbb{M}_{n\times n}(\mathbb{R}) \to \mathbb{M}_{n\times n}(\mathbb{R})$ the linear operator such that $T(M)=M^t$, where $M^t$ is the transpose of the matrix $M$. Prove that $T$ is ...
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1answer
591 views

Covariance matrix of two matrices- how to calculate

In maximum covariance analysis, to extract correlated columns, it is asked to calculate the covariance matrix. For two vectors, corvariance matrix is understood, COV(v1,v2) = v1*v2' How do I ...
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3answers
452 views

Question about determinants

I am working on some practice problems and I'm unsure where to begin this problem. It starts off by giving $\det(X)= 1$ for the following matrix $X$:$$ \begin{matrix} a & 1 & d \\ b & 1 ...
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1answer
511 views

Why does SVD provide the least squares solution to $Ax=b$?

I am studying the Singular Value Decomposition and its properties. It is widely used in order to solve equations of the form $Ax=b$. I have seen the following: When we have the equation system $Ax=b$, ...
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2answers
159 views

Prove that $\det(M-I)=0$ if $\det(M)=1$ and $MM^T=I$

$M$ is a $3 \times 3$ matrix such that $\det(M)=1$ and $MM^T=I$, where $I$ is the identity matrix. Prove that $\det(M-I)=0$ I tried to take $M$ $=$ $$ \begin{pmatrix} a &b & c \\ ...
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3answers
2k views

Closest Point to a vector in a subspace

Given v = [0 -3 -8 3], find the closest point to v in the subspace W spanned by [6 6 6 -1] and [6 5 -1 60]. This is web homework problem and I have used the formula (DotProduct(v, w.1)/DotProduct(w.1, ...
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4answers
748 views

Finding nonnegative solutions to an underdetermined linear system

Here's the environment of my problem: I have a linear system of 4 equations in 8 unknowns (i.e. $Ax = b$, where $A$ is $4 \times 8$, $x$ is $8 \times 1$, and $b$ is $4 \times 1$, with $A$ given and ...
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1answer
176 views

Condition number vs. reconstruction error

Suppose I want to solve a simple, linear inverse problem given by $\mathbf{y} = \mathbf{A} \cdot \mathbf{c}$ where $\mathbf{A}$ is an $M \times K$ matrix and I want to solve for $\mathbf{c}$ ($M$ = ...
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19answers
15k views

What is the difference between a point and a vector

I understand that a vector has direction and magnitude whereas a point doesn't. However, the course note that I am using states that a point is the same as a vector. Also, can you do cross product ...
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1answer
194 views

Functional Analysis - Finding the multiplicative operator norm over L1

Let $f \in C([0,1])$, the space of continuous real-valued functions over $[0,1]$. Let $\Gamma_f: L^1([0,1],m) \rightarrow L^1([0,1],m)$, the space of complex-valued functions Lebesgue integrable ...
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3answers
695 views

Why a non-diagonalizable matrix can be approximated by an infinite sequence of diagonalizable matrices?

It is known that any non-diagonalizable matrix, $A$, can be approximated by a set of diagonalizable matrices, e.g. $A \simeq \lim_{k \rightarrow \infty} A_k$. Why this is true? Note: I was faced with ...
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1answer
567 views

Show that $\{1, \sqrt{2}, \sqrt{3}\}$ is linearly independent over $\mathbb{Q}$.

My apologies if this question has been asked before, but a quick search gave no results. This is not homework, but I would just like a hint please. The question asks Show that $\{1, \sqrt{2}, ...
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3k views

How to prove invertibility of a linear independent column matrix?

How it can be proved that a matrix whose columns are linearly independent such as a basis matrix that spans a space is invertible?
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Intuition behind Matrix being invertible iff determinant is non-zero

I have been wondering about this question since I was in school. How can one number tell so much about the whole matrix being invertible or not? I know the proof of this statement now. But I would ...
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1answer
6k views

Calculating the number of operations in matrix multiplication

Is there a formula to calculate the number of multiplications that take place when multiplying 2 matrices? For example $$\begin{pmatrix}1&2\\3&4\end{pmatrix} \times ...
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1answer
154 views

optimizing a logdet function with respect to a scalar and the Hessian matrix

Given a logdet function $\mathcal{L}(\gamma)$, $$ \mathcal{L}(\gamma) = \log\vert \mathbf{I} + \gamma\mathbf{S} \vert - \mathbf{q}^T(\gamma^{-1}\mathbf{I} + \mathbf{S})^{-1} \mathbf{q}, $$ where ...
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1answer
2k views

Proving replacement theorem?

I want to see if I am understanding the proof of the replacement theorem correctly. Let $V$ be a vector space that is spanned by a set $G$ containing $n$ vectors. Let $L \subseteq V$ be a linearly ...
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1answer
106 views

What is an overlap?

I want to ask what an overlap is. My teacher said that for example $1$: Everything is an overlap hence it is not locally finite. For example $2$, it doesnt overlap. Please teach me these two ...
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142 views

An open cover that is not locally finite

I could not understand why example $13.4$ is not locally finite. Can you give me an explanation please.
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Properties of the Cone of Positive Semidefinite Matrices

The set of positive semidefinite symmetric real matrices form a cone. We can define an order over the set of matrices by saying $X\geq Y$ if and only if $X-Y$ is positive semidefinite. I suspect that ...
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1answer
5k views

Hermitian Matrices are Diagonalizable

I am trying to prove that Hermitian Matrices are diagonalizable. I have already proven that Hermitian Matrices have real roots and any two eigenvectors associated with two distinct eigen values are ...