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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Is there a fundamental meaning of kernel in “kernel function” and “kernel of linear map”

In pattern analysis kernel trick is famous, based on kernel function. On the other hand kernel of linear map is the null space. Is there a deep relation between this two "kernel" words or there is no ...
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18 views

Why does the Gaussian-Jordan elimination works when finding the inverse matrix?

In order to find the inverse matrix $A^{-1}$, one can apply Gaussian-Jordan elimination to the augmented matrix $$(A \mid I)$$ to obtain $$(I \mid C),$$ where $C$ is indeed $A^{-1}$. However, I fail ...
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3answers
31 views

Does matrix has a underlying basis?

Can I say a matrix (M) as a liner transformation and it operates on a vector? The vector must have a basis and the matrix M gave us a new vector. Now is there any basis associated with the matrix. ...
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10 views

Infinite subspaces for a vector space that cannot be spanned by a single element

If a vector space (over an infinite field) cannot be spanned solely by a single element, does it mean it has infinite subspaces? I couldn't find an example that contradicts this
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2answers
39 views

Show that there exists a $3 × 3$ invertible matrix $M$ with entries in $\mathbb{Z}/2\mathbb{Z}$ such that $M^7 = I_3$.

Show that there exists a $3 × 3$ invertible matrix $M$ (which is not the identity matrix) with entries in the field $\mathbb{Z}/2\mathbb{Z}$ such that $M^7 = $Identity matrix. All I could do was use ...
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1answer
15 views

Cholesky factorization exist?

Is there a theorem or a way to show that if I have a real and symmetric positive definite matrix $A$ and its Cholesky factorization is $A = LL^T$ then $B = L^TL$ is also positive definite? Or in other ...
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1answer
34 views

Linear algebra references for a deeper understanding of quantum mechanics

I'm a graduate student studying quantum mechanics/quantum information and would like to consolidate my understanding of linear algebra. What are some good math books for that purpose?
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19 views

Spectral Radius of a Block Matrix

I have real matrix $P$ obtained from numerical solution (FEM) of a physical problem, as \begin{equation} P=P_1+P_2= \begin{bmatrix} A_{2n \times 2n}&B_{2n \times n}\\C_{n \times 2n}&D_{n ...
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9 views

Proof of $\mathcal{R}(A^+)=\mathcal{N}(A)^\perp$ where $A^+$ is the MP pseudoinverse, is it correct?

Yesterday, I asked if that property was true : Rangespace of Moore-Penrose pseudoinverse : $\mathcal{R}(A^+)=\mathcal{N}(A)^\perp$? Someone found a demonstration, but he used SVD, and I wanted to ...
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2answers
20 views

Proving $dim(img(g\circ f)) \leq dim(im(f))$

I want to prove: $dim(img(g\circ f)) \leq dim(im(f))$. $g$ and $f$ are linear maps. The map $f$ first maps the input $x$ into $im(f)$. $g$ take $im(f)$ into $im(g)$. Thus, this last map is ...
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1answer
18 views

How transformation of co-ordinates system relates to its vectors?

Consider a positive definite matrix. Can we consider that it has a underlying co-ordiante system? If we transform that co-ordinate system how the the vectors are transformed? Is this question even ...
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90 views

Show that if $AA^t = A^tA$, then $A=A^t$

Suppose $A$ is a matrix with non-negative real entries. If $A^tA = AA^t$, show that $A=A^t$. My proof says: $AA^t = A^tA = (AA^t)^t$. I can't seem to get to the point of $A=A^t$ Edit: What if $A$ is ...
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1answer
14 views

Meaning of co-ordinate system of Covariance matrix

Can we think that any matrix representation has an underlying co-ordinate system? Now consider a positive definite sample covariance matrix. If so what is the meaning of the co-ordinate system of the ...
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3answers
40 views

Find the adjoint

Choose one from he following list of inner products and then find the adjoint of: $$ \left[ \begin{array}{ c c } 1 & 2 \\ -1 & 3 \end{array} \right] $$ When your inner prod cut ...
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16 views

Diagonal matrices in ${\rm SL}(2, \mathbb{K})$

How can one using first order logic distinguish diagonal matrices in ${\rm PSL}(2, \mathbb{K})$ in some basis? I'm trying to do it as follows: Distinguishing maximal number of commuting ...
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1answer
37 views

How to prove the equality of two matrix expressions

I am new to linear algebra and my question maybe too simple. I have a n-by-m matrix $D$ that its columns have unit L2 norm. Let $D_a$ be a sub-matrix of $D$ composed by some columns of $D$. I need to ...
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1answer
19 views

Linear space obtained from another one factoring out the constant.

Given a three dimensional vector space $H$, I don't understand what is the two dimensional vector space obtained from $H$ by factoring out the constants. Someone can explain me that? Thanks!
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1answer
28 views

$M_n$ be space of compex n x n matrices with inner product $(A,B)= Tr A\bar{B}^t$

Let $M_n$ be space of compex n x n matrices with inner product $(A,B)= Tr A\bar{B}^t$. Find adjoint operators: i. For operator of left multiplication $L_A :X\rightarrow AX$ by matrix A for $X \in ...
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1answer
26 views

Rank of matrices, prove inequality [duplicate]

Today I'm having hard time with linear algebra problems; this is one: $\forall A,B\in M_n(\mathbb{K})$, $\mathrm{rank}(A)+\mathrm{rank}(B)\le \mathrm{rank}(AB)+n$ $M_n(\mathbb{K})$ is the space of ...
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1answer
33 views

Let P be a space of real polynomials of degree $\le 2$ with inner product [on hold]

Let $P$ be a space of real polynomials of degree $\le 2$ with inner product $\langle p,q \rangle = \int_{0}^{1}p(x)q(x)dx$ . Find volume of the set of polynomials $a+bx+cx^2$ with coefficients ...
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1answer
46 views

Coefficient calculation on Fourier series !?

in a Fourier series for function $$f(x)=\begin{cases}-1&\text{for }-\pi<x<0\\\sin x&\text{for }0<x<\pi\end{cases}$$ with $f(x)=f(x+ 2 \pi)$, is $f(x)= \dfrac{a_0}{2}+ ...
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25 views

Is it possible to have a matrix with eigenvalues that cannot be constructed from a finite number of basic arithmetic operations, and nth roots?

For example, a characteristic polynomial $ p(\lambda) = \lambda^5 - \lambda -1 $ has the root 1.167304..., but this number cannot be written as a finite number of arithmetic operations (addition, ...
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31 views

Does there exist a steady state vector of this Markov Matrix?

Does there exist a steady state vector of Markov Matrix $$P=\begin{bmatrix} \frac{1}{2} & \frac{1}{3}\\ \frac{1}{2} & \frac{2}{3} \end{bmatrix}$$ Initially I was not sure whether to answer ...
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1answer
21 views

questions on polynomial Lagrange Interpolation of order $n$?

I ran in One Ex in my book when I‌ prepare for final exam on numerical method. how can help me how we solve such a problem? if $P(x)$ and $Q(x)$ be two polynomial Lagrange Interpolation of order $n$ ...
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1answer
30 views

Verifying linear independence for $\left\{ \sqrt{x} , \sqrt{x+1}, \sqrt{x+2} \right\}$ using Wronskian

I need to verify the linear independence for the group of functions: $$\left\{ \;f_1 = \sqrt{x} , \;f_2 = \sqrt{x+1}, \;f_3 = \sqrt{x+2} \right\}$$ using Wronskian, for $x > 0$. I wasn't told ...
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1answer
22 views

Method of moments estimation for $\theta$

I read one example in my notes, but I couldn't find out how the answer in my notes is derived. If $x_1,...,x_n$ are realizations of a random variable distributed with the following PDF: $f(z; ...
2
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1answer
51 views

Largest eigenvalues of AA' and A'A [on hold]

Prove that for every real matrix $A$, the largest eigenvalue of $A'A$ equals the largest eigenvalue of $AA'$ (where ' means transpose). Thanks!
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1answer
23 views

Eigenvectors and geometrical transformation

$$A= \begin{pmatrix} 2/3 & 2/3 & -1/3 \\ 2/3 & -1/3 & 2/3 \\ -1/3 & 2/3 & 2/3 \\ \end{pmatrix}$$, I need to understand that kind of ...
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8 views

Checking reasoning for finding Jordan-Canonical Form for 3x3 matrix

$$A=\left[\begin{array}{ccc}2 & 2 & 3 \\1 & 3 & 3 \\-1 & -2 & -2\end{array}\right]$$. The eigenvalues of a (including the multiplicity) = 1,1,1. $$A-I= ...
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1answer
30 views

Shortest distance proof

Show that the shortest distance from a point $P$ to the line through $P_0$ with direction vector $\overrightarrow{d}$ is $$\frac{|\overrightarrow{P_0P}\times ...
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0answers
33 views

Existence of a vector in a given basis of a vector space with increasing coordinates

Is it true that $$ \forall\, x=(x_1,x_2,\ldots,x_n)\in\mathbb{R}^n \,\,\, \forall\, y=(y_1,y_2,\ldots,y_n)\in\mathbb{R}^n \,\,\, \exists\, \alpha\in\mathbb{R} \,\,\, \exists\, \beta\in\mathbb{R} \,\, ...
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3answers
59 views

If a matrix A square is 0, does it follow that A = 0? [duplicate]

Let A be a square matrix. If $A^2 = 0$, then it follows that $A = 0$. Is there a counterexample for this? If there isn't, what kinds of explanation can I make to justify this statement?
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The relation of determinants between linear transformation.

I am studying the simplicity of PSL and came across the above statement. I don't understand why $\det L_c =c \det L$? (Given two set of basis of the same vectorspace, $v_1,...v_n$ and $w_1,...w_n$, ...
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0answers
10 views

there is a unitary $U \in {M_m}$ such that $X = YU$. Why $X$ and $Y$ have the same range?

Let $X,Y \in {M_{n \times m}}$ have orthonormal column. Also there is a unitary $U \in {M_m}$ such that $X = YU$. Why $X$ and $Y$ have the same range(column space) ?
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71 views

How to prove this condition of inner product

I want to prove this condition: $\langle f(x),f(x)\rangle >0$ if $f(x) \neq 0$. Given $\langle f(x),f(x) \rangle = \int_{-1}^1 (x^2)(f(x))^2 dx$ Anyone can help?thank you!
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0answers
21 views

What is the difference between the middle factor and the middle term of permutation ? [duplicate]

What is the difference between the middle factor and the middle term of permutation ?
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13 views

Triangulation of matrices

Suppose that $A$ is some triangularizable matrix in $M_n(\mathbb R)$. The usual approach I know of to find a triangular matrix similar to it is to find bases for all the eigenspaces, then find their ...
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25 views

Let A be 10x10 matrix such that A^2 =0 [on hold]

Show that the Column Space of A is a subset of the Null Space of A. Show that the Rank(A) must be less than or equal to 5.
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1answer
28 views

Eigenvector and Eigenvalues of a square matrix.

Define what is meant by saying that v is an eigenvector with associated eigenvalue λ for the square matrix A. Just a definition question that I was hoping to get help with. It's from a past exam ...
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2answers
29 views

prove that if $S$ is a subspace of $V$,then $S ∩ S^\perp =\{0\}$

Anyone can help me with this? I have to show that the intersection between $S$ and orthogonal complement of $S$ is $\{0\}$.
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1answer
49 views

Solving double angle trigonometric equities [on hold]

I am having problems understanding how to solve the equation: $\sin{(2x)} = \sin{(x)}\cos{(x)}^{1/2}$
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2answers
26 views

How to get a linear plot of a power function?

Imagine I have a function of x as follows: $$y=f(x) = ax^2 + bx + c + \frac{d}{x}$$ And I am trying to plot this on a graph with y as ordinate and $x^{n}$ as abscissa. Now what value of n would give ...
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40 views

Why is unit circle not sufficient to bound the partial sums? [on hold]

I want to find vectors $\textbf{v}_1, \dots,\textbf{v}_n$ in $\mathbb{R}^2$ with that $\sum_{i=1}^n\textbf{v}_i=\textbf{0}$ and $\Vert \textbf{v}_i\Vert\leq 1$ for all $i=1,\dots,n$, such that for ...
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34 views

question asks when is the birthday??? [duplicate]

Question asks how to find out Cheryl's birthday??
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2answers
23 views

Rearranging into $y=mx+c$ format and finding unknowns $a$ and $b$

Two quantities $x$ and $y$ are connected by a law $y = \frac{a}{1-bx^2}$ where $a$ and $b$ are constants. Experimental values of $x$ and $y$ are given in the table below: $$ ...
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inequalities for $tr(AB)$ , where A and B symmetry, positively definite matrix

Let $A$ and $B$ be two symmetry, positively definite $n\times n$ matrix with positive eigenvalue $a_1,...,a_n$ and $b_1,...,b_n$ respectively. What's the relationship between them and $tr(AB)$? Are ...
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1answer
11 views

$X$ and $Y$ have the same range(column space). Why there is a unitary $U \in {M_m}$ such that $X = YU$?

Let $X,Y \in {M_{n*m}}$ have orthonormal column. Also $X$ and $Y$ have the same range(column space). Why there is a unitary $U \in {M_m}$ such that $X = YU$?
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2answers
35 views

Prove an eigenvector for two matrices is also the eigenvector for the product of those matrices. [duplicate]

So let's assume that A and B are both nxn matrices, and that u is an eigenvector for both A corresponding to lambda one and B corresponding to lambda 2. I need to prove that u is also the eigenvector ...
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17 views

Question related to image of $[1,N]^n$ under a linear tranformation

I am reading an article and I am a bit confused about the following passage. I would appreciate any clarification. It goes as follows: Let $\bar{F}$ be a collection of $r$ linearly independent ...
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20 views

direct sum of modules is isomorphic to the direct sum permuting indices? [on hold]

the primary for the exercise idea is to use the universal property of the external direct sum of modules.