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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Prove that $\text{det}(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$

Let $f(x)=(p_1-x)\cdots (p_n-x)$ $p_1,...p_n\in \mathbb R$ and let $a,b\in \mathbb R$ such that $a\neq b$ Prove that $\text{det} A={bf(a)-af(b)\over b-a}$ where $A$ is the matrix: ...
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33 views

Prove that $\lambda_1^2$, $\lambda_1\lambda_2$ and $\lambda_2^2$ are eigenvalues of matrix $A$

This is the problem I am currently having trouble with: If $\lambda_1$ and $\lambda_2$ are eigenvalues of matrix $$ \begin{bmatrix} a & b\\ c & d\\ ...
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1answer
25 views

Vector norm lemma and proof

I have a question from Numerical linear algebra book by Trefethen & Bau : Let $\|\cdot\|$ denote any norm on $C^m$. The corresponding dual norm $\|\cdot\|'$ is defined by the formula ...
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Prove that $\text{rank } T = \operatorname{rank} T^2 \iff \operatorname{Im}T \cap \ker T = \{ \vec 0\}$

$\newcommand{\r}{ \operatorname{rank} } $ Let $T: V\to V$ be a linear transformation with $\dim V< \infty$. Prove that: $$ \r T = \r T^2 \iff \operatorname{Im} T \cap \ker T = \{ \vec 0 \}.$$ ...
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Proof of non-singularity given certain conditions

Suppose that I have a $n\times t$ matrix $\boldsymbol{X}$ that is full rank and a non-singular matrix $\boldsymbol{L} = \begin{bmatrix} \boldsymbol{L}_1 & \boldsymbol{L}_2 \end{bmatrix}$ such that ...
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24 views

[Part of a system of linear equations!]: Find $B$ such that $A = B\times C$, but $C\times C'$ is non-invertable

I have the following Equation: $A = B\times C$ $A$ is a $(N\times 1)$ Known Matrix $B$ is a $(N\times M)$ Unknown Matrix, where $N>M$ $C$ is a $(M\times 1)$ Known Matrix $C\times C'$ is a ...
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23 views

Moving vectors to the left and the right of a product

Suppose that $A$ and $B$ are $1\times n$ row vectors and $x$ is a $n\times 1$ column vector. I have an expression $$ (Ax)^2B'B $$ which is an $n\times n$ matrix. Question: Is it possible to write ...
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22 views

Showing a set of vectors is independent

I'd like to redo the proof by walkar (with fewer vectors) just to see if I can handle the algebra the way I do Proving that $(u+v,u+w,v+w)$ is linearly independent Suppose $au + bv = 0 \implies a = ...
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36 views

For a linear function the following are equivalent: continuity and Lipschitz continuity

Let $(X,||\cdot ||_X)$ and $(Y,||\cdot ||_Y)$ be normed Vectorspaces over a common field $\Bbb K$. Let $A:X \to Y$ be a linear function. I have to show that the following statements are equivalent: ...
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Finding altitude and azimuth with an accelerometer and magnetometer

I posted this in the astronomy stack exchange forum, but considering that it is a very math intensive question I figured there could also be people on here that could help. For a project with my ...
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46 views

If dim V = dim U and $S \circ T$ is onto, prove or disprove V and W are isomorphic

Let $T: \mathbb V \to \mathbb W$ and $S: \mathbb W \to \mathbb U$ be linear maps. If dim $\mathbb V$= dim $\mathbb U$ and (S o T) is onto (composition), then $\mathbb V$ and $\mathbb W$ are ...
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12 views

Prove that eigenvalues of the operator lie in the interval [on hold]

Let $\phi$ and $\psi$ be two self-adjoint linear operator in Euclidean space, the eigenvalues of which lie respectively in the intervals $[c; d]$ and $[m; n]$. Prove that eigenvalues of the operator ...
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Subtracting scaled projection matrix from identity matrix

I am trying to understand what the following operation signifies. $$ \rm W_n=I-2u_n u_n^H/u_n^Hu_n $$ where I and $u_n$ is described in section 6.3.4.2.3 of this document. My question is, what does ...
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Let $S$ be a set of vectors. If each finite subset of $S$ is linearly independent, then $S$ is linearly independent

Here's a similar question which doesn't answer mine: If every subset of $S$ is linearly independent, then $S$ is independent Let $S$ be finite. Let $S_1 \cup S_2 \cup \ldots \cup S_n = S$ such that ...
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1answer
107 views

Eigenvectors of “weighted” Hermitian matrix?

Consider two real matrices $\boldsymbol{H}$ and $\boldsymbol{D}$ with the following properties: $\boldsymbol{H}$ is a symmetric matrix (since it is a real matrix this is equivalent to being ...
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80 views

Can a elementary row operation change the size of a matrix?

My question is very simple - Can an elementary row operation change the size (eg: $2\times2$ or $3\times 2$) of a matrix? I think the answer should be no, but while reading Linear Algebra by Hoffman ...
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1answer
54 views

Lie bracket of $\mathfrak{so}(3)$

I know that for $\mathfrak{so}(3)=\mathcal{L}(SO(3))$, the set of $3\times 3$ real antisymmetric matrices, we can define a basis $$T^1=\begin{pmatrix}0&0&0\\ 0&0&-1\\ ...
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1answer
29 views

Systems of linear equations in the same modulus

I am working with a system of linear equations all taken with the same modulus, $n$, there is no assumption on $n$ other then it is at least 3 (really don't want to assume it is prime) I don't have ...
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56 views

Express eigenvectors of $A^{-1}$ in terms of eigenvectors of $A$

I know the eigenvalues of the matrix $A^{-1}$ are $\frac{1}{\lambda_n}$ where $\lambda_n$ are the eigenvalues of $A$. I didn't know their eigenvectors were related; in what way are they related? Also ...
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1answer
44 views

Degrees of freedom in a $n \times n$ table

Suppose we have an $n \times n$ table where each row and each column sums to some number $k$. Say that the elements of the table and $k$ are real numbers. Now the question is how many places can we ...
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finding inner product

This is from my textbook: I don't know how to tell whether the spanning set are actually orthogonal. The textbook's solution is like this, forexample, to see if $P_0(t)$ and $P_1(t)$ are orthognal, ...
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Interval bounds for symmetric doubly-stochastic matrices (designed with Metropolis weights).

I'm facing an unusual problem with doubly-stochastic matrices, in the context of some undirected graph. I assume that it is connected, but this is not so important for this problem. Let me introduce ...
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Elementary reflector $Q$ is orthogonal iff

Recall that an elementary reflector has the form $Q = I + \alpha xx^T\in\mathbb{R}^{n\times n}$ with $\|x\|_{2}\neq 0$. Show that $Q$ is orthogonal iff $$\alpha = \frac{-2}{x^Tx} \ \ \text{or} \ \ ...
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519 views

Can a matrix satisfy all three of the following properties?

Consider an $n \times n$ matrix of the form $$ A = \begin{bmatrix} a_1 & a_2 & \ldots & a_{n-1} & a_n \\ 1 \\ & 1 \\ & & \ddots \\ & & & 1 \end{bmatrix} $$ for ...
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36 views

Irreducible polynomials in $\mathbb F_3[x]$

Finding irreducible polynomials in $\mathbb F_3[x]$ of degree less or equal to $4$ for $d=2,3$ the polynomial should not have a root case $d=2$ there are $2\cdot 3\cdot 3=18$ polynomials with ...
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42 views

How to recover a $K$-algebra from endomorphism algebra of forgetful functor?

I'm trying to work out how a finite-dimensional $K$-algebra $B$ can be recovered from the algebra of endomorphisms of the forgetful functor $\omega$ from the category of finitely generated left ...
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Is a subset of a linearly independent set in a vector space linearly independent?

The empty set is linearly independent. Also, $S_1$ is linearly independent if $S_1 = S_2.$ Suppose $v_1, \ldots, v_n \in S_2.$ Then $a_i = 0$ for all $i$ in $a_1v_1 + \ldots + a_nv_n = 0.$ Now ...
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58 views

Square matrix over $\mathbb{Z}$ can't have $\frac{1}{4}(-3+ i \sqrt5)$ as an eigenvalue

Prove square matrix over $\mathbb{Z}$ can't have $\frac{1}{4}(-3+ i \sqrt5)$ as an eigenvalue. My proof: If matrix has eigenvalue z=$\frac{1}{4}(-3+ i \sqrt5)$, then it must has eigenvalue ...
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1answer
26 views

Prove that the dimension of row space equals to the dimension of column space of an $n\times n$ matrix

Knowing that the row space of $A\in \mathbb{R}^{n\times n}$ equals $N(A)^\perp$ prove that the dimension of column space of a matrix equals its row space dimension. So I'm trying to apply ...
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How many ordered basis does $V$ have? [duplicate]

If $F$ is a field with $q$ elements, $V$ a $F$-vector space of dimension $n$, then how many ordered basis does $V$ have ? First, $V$ has $q^n$ elements, am I correct ? Let $\{v_1,\dots,v_n\}$ be ...
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Finding bases for the kernel and image of $M= \begin{bmatrix}1&1&1\\-s&2+s&-1\\-s&s&1\end{bmatrix}$

So I have a Matrix $$M= \begin{bmatrix}1&1&1\\-s&2+s&-1\\-s&s&1\end{bmatrix}$$ and a linear transformation $$f:\mathbb{R}^3 \to \mathbb{R}^3, \quad x \mapsto Mx$$ and I'm ...
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27 views

Norm of a dot product?

I am reading a paper and is rather perplexed by the following equation. Particularly, inside the double bar || which I believe is the norm, there is a dot product. If that is the case, what does it ...
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3answers
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How can I prove: $\forall \lambda \in C$ with $|\lambda| = 1$, exist unitarian matrix $B$ with eigenvalue $\lambda$

How can I prove: $\forall \lambda \in C$ with $|\lambda| = 1$, exist unitarian matrix $B$ with eigenvalue $\lambda$. I tried to find a counter-example and I was not succeeded. I believe I need to ...
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1answer
25 views

Find spectral theorem of $A$ and find its singular eigenvalues.

The rotation matrix $$A=\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$ has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and ...
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1answer
19 views

Given $f(x)=x^2-4x+3$, find the points on the curve $y=f(x)$ where the tangent to the curve passes through -6.

Given $f(x)=x^2-4x+3$, find the points on the curve $y=f(x)$ where the tangent to the curve passes through $(0,-6)$. State the equations of the tangents at these points. Hi everyone, I tried to find ...
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Division vector space over binary extension

I'm working in the maths behind the Rijndael's algorithm and I get stalled with the mixColumns() operation. I'm writing code to do the algorithms without pre-calculation tables (the objective is to ...
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1answer
35 views

Existence of Unimodular Congruence Transformation for Symmetric, Integer matrices

Two symmetric, integer valued matrices, $K_1$ and $K_2$, are congruent if there exists a unimodular integer matrix, $X$, such that $$X^T K_1 X = K_2$$ What are the conditions on the existence of such ...
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Rank of a symmetric matrix after removing a column and row.

If I have a $n\times n$ symmetric matrix $M$ with real entries, zeros on the diagonal, and two of the column vectors are identical and I remove one of these columns, and the corresponding row, then ...
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Query about the Moore Penrose pseudoinverse method

I have recently discovered the Moore-Penrose psuedoinverse method, and I am currently testing the waters with it. I noticed if I have a system, say $$a_1x_1=0$$ $$a_2x_1+a_3x_2=0$$ $$\vdots$$ ...
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1answer
34 views

Effect of simple linear transformation

Consider the linear transformation given by $$T\left\{\begin{bmatrix}x \\y\\z\end{bmatrix}\right\} =\begin{bmatrix}-x\\y\\z\end{bmatrix}$$ Find a matrix $A$ such that $T(x) = Ax$, where x = $[x, y, ...
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Eigenvectors of Generalized Sylvester Equation $AX+XB^\text{T}=\lambda CXD^\text{T}$

Ok here's what I mean with the Sylvester equation eigenvectors. The simplest case, where $C = D = I$, has already been solved in the literature (Matrix Calculus by W.H. Steeb). $$A X + X ...
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PTM using Hastings-metropolis [on hold]

[Compute the 4 × 4 PTM (pij ) under the T = 2 dynamics of Hastings–Metropolis][1]
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2answers
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Find the possible equations of planes containing a line

If I had a line in $R^3$ that had all its points described by $P =\begin{pmatrix} a\\ b\\ c \end{pmatrix}+\lambda \begin{pmatrix} x\\ y\\z \end{pmatrix}$ where $a, b,$ and $c$ are constants, what ...
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Does $B^2 \leq A^2$ imply $\| A^{-1} B\| \leq 1$ for the operator norm?

Assume we have two $n \times n$ real symmetric matrices $ A^2 $ and $B^2$, such that it holds for some $0\leq\rho<1$ $$ 0 < (1-\rho)B^2 \leq A^2 \leq (1+\rho)B^2, $$ where "$\leq$" means ...
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18 views

Number of vectors in a span over a certain field

Let $S = \{u_1, u_2,\ldots, u_n\}$ be linearly independent subset of vector space $V$ over $\mathbb Z_2$. Number of vectors in $\operatorname{span}(S)$? Consider $u_1 + u_2 + \cdots + u_n \in ...
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Is there any linear algebra textbook presented using logical symbols?

I'm currently going through a book called Linear Algebra Done Right by Axler, and to be honest, his book seems to be very loose with what things he defines. For instance , the symbol 0 could be mean a ...
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148 views
+100

Given a symmetric positive-definite matrix $M$, find all $A$ such that $A^\top M A=M$

Given $M$ a real symmetric positive-definite matrix, I would like to characterise all matrices $A$ such that $A^\top M A=M$. Note that the question of finding $A$ solutions to $A^\top M A=M$ for all ...
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1answer
34 views

Proof about orthogonality of columns of a matrix

Consider a matrix $A \in \mathbb{R}^{n \times n}$ and the canonical inner product in $\mathbb{R}^{n}$. Show that if the rows of A form an orthogonal set, the same happens with the columns. So ...
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0answers
15 views

Finite prime field representation of uniform matroid $U_{2,n}$

Suppose I have a uniform matroid $U_{2,n} = (E, I)$ (so $F \subset E$ has $F \in I \iff |F| \leq 2$) and want to represent it over $GF(p)$, i.e. I would like to construct a map $\phi : E \to GF(p)^2$ ...
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2answers
32 views

How do I specify the inverse of a correlation matrix?

To specify a correlation matrix $\in \mathbf{R}^{n\times n}$. There are $n(n-1)/2$ free elements. If I wanted to specify a matrix that is the inverse of some correlation matrix, how should I specify ...