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[Vector spaces ]Why does the matrix of a linear transformation from $V$ ( of dim $n$) to $W$ ( of dim $p$) have $p$ rows?

As others have already commented, it's because $W$ is $p$-dimensional, so each image vector $f(b_i)$ has $p$ coordinates with respect to the basis $c_1,\ldots,c_p$, and therefore $p$ rows in the ...
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Construction of the matrix of a linear map between two vector spaces : difficulty in understanding how an expression represents a coordinate vector.

If you accept that the $[F(v_i)]_{B_W}$ are column vectors (one for every i from $1$ to $n$) then the sigma expression is just a sum of scaled column vectors, i.e. a column vector.
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Is Moore–Penrose inverse a synonym for pseudoinverse?

As stated in the references, the "pseudoinverse" is also called Moore–Penrose inverse. Let us define $\mathbf{A} \in \mathbf{R}^{m\times n}$. Strictly speaking, the matrix $\mathbf{B} \in \...
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For $A\in \mathbb R^{n\times n}$, when is $A+A^T$ positive semi-definite?

This seems equivalent to the condition of $A$ being an "accretive operator" (from this paper) Looking at the action of $B$ from the example above, you can see that for quite a few vectors ...
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Exercise 10, Section 3.4 of Hoffman’s Linear Algebra

Expanding on a way from my hint: You noted if $S \neq 0$ and $S \neq I$, we must have vectors $\alpha_1, \alpha_2 \in \mathbb{R}^2$ with $S \alpha_1 \neq 0$ and $S \alpha_2 \neq \alpha_2$. To satisfy $...
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Why does the Gaussian-Jordan elimination works when finding the inverse matrix?

Let me make it concrete in the following example(@pauly-b has written a much more beautiful answer above). Suppose that 4 apples(any two are the same price) and 3 bananas(any two are the same price) ...
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Determine axis of rotation from 4x4 transformation matrix

If you are using a 4x4 homogeneous transformation matrix, the point is either located in the last column or the bottom row, depending on conventions. Note that the element at 4,4 is typically 1 and ...
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Determinant of $12\times12$ matrix

Your proof is correct. Regarding your second question, let $\mathbf{1}$ be a vector of all ones. Then $ A = \mathbf{1}\mathbf{1}^T-I. $ Now use Cauchy's determinant formula $$ \det(\mathbf{1}\mathbf{1}...
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Prove that an upper triangular matrix $A$, such that $A^*A = AA^*$, must be diagonal.

Partition $A$ as $$ A = \begin{pmatrix} A_{11} & A_{12}\\0 & A_{22}\end{pmatrix}. $$ If $A$ is normal, then $$ AA^* = \begin{pmatrix} A_{11}A_{11}^* + A_{12}A_{12}^* & \star\\\star & \...
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Show that $A \cdot \textbf{0}_{n \times p}=\textbf{0}_{m \times p}$

$A \cdot \textbf{0}_{n \times p}=\textbf{0}_{m \times p}$ For all $v\neq 0$ $A\cdot \textbf{0}_{n \times p} v =0$ and then use the uniqueness of the zero element.
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Show that $A \cdot \textbf{0}_{n \times p}=\textbf{0}_{m \times p}$

Let us use $0$ to mean either the LHS Zero Matrix or the RHS Zero Matrix, with Correct Size. $A(0) = A(0+0) = A(0)+A(0)$ [[ This is because 0+0=0 & Multiplication is Distributive over Addition ]] $...
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Show that $A \cdot \textbf{0}_{n \times p}=\textbf{0}_{m \times p}$

If you're allowed to use the algebraic properties of matrices, then observe that: $$A0 = A(0+0) = A0 + A0$$ Since $\mathbb{F}^{r \times s}$ is a vector space, an additive inverse $-(A0)$ exists. Hence:...
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Solving recurrence relation $a_n = a_{n-1} - a_{n-2}$

For $$A_n=A_{n-1}-A_{n-2}.....(1)$$ Take $A_n=t^n$, then we get $t^2-t+1=0 \implies t=-w,-w^2$ where $w=e^{2in\pi/3}, w^2=e^{-2in\pi/3}$ are cube-root of unity. So the solution of (1) $$A_n=p(-w)^n+q(...
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Solving recurrence relation $a_n = a_{n-1} - a_{n-2}$

You are asking for the recurrence, I think, $a_0=a_1=1$, $a_n=a_{n-1}-a_{n-2}$. Let $T:\Bbb R^3\to\Bbb R^3$ be the linear operator with the following matrix (on the standard basis): $$\begin{pmatrix}1&...
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What´s the normal form of the generator matrix?

In $\mathbb{Z}_5$, we have $3 = -2$ since they differ by $5$, and similarly $4 = -1$. (For some reason, you're looking at residues modulo $6$ in the arithmetic in your last line.) The first row has $(-...
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What is the matrix corresponding to the quadratic form $x^2 + y^2 + z^2$?

When you want to find the matrix of a quadratic form, as with linear maps you merely need to evaluate it at basis vectors. But for quadratic forms you need to evaluate it at pairs of basis vectors. In ...
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What is the matrix corresponding to the quadratic form $x^2 + y^2 + z^2$?

$$ A = \begin{pmatrix}1 & 0 & 0 \\ 0& 1&0 \\ 0 & 0 & 1\end{pmatrix}$$
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Derivative of determinant of square matrix-valued function

That's just the usual product rule. In two dimensions, $$ \begin{align} {d\over dt} \left(a(t)d(t)-b(t)c(t)\right) &=a'd-b'c+ad'-bc'\\ &= \begin{vmatrix} a' & b'\\ c & d \\ \end{...
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2 votes

Taylor Expansion of a function that maps matrices to scalars

Basing on the fact that $\mathbb{R}^{n\times m}\simeq \mathbb{R}^{nm}$ we get for $X=\{x_{ij}\}$ $$\displaylines{f(\tilde{X})\approx f(X)+\sum_{i=1}^n\sum_{j=1}^m{\partial f\over \partial x_{ij}}(X)\,(...
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Circulant-like determinant

Let $\zeta\in\mathbb{C}$ with $\zeta^{2n}=1=-\zeta^n$. I think, $v=(1,\zeta,\ldots,\zeta^{n-1})$ is an eigenvector with eigenvalue $\sum_{k=0}^{n-1}a_k\zeta^k$. Now let $\zeta:=e^{\pi i/n}$. This ...
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4 votes

Exercise 10, Section 3.4 of Hoffman’s Linear Algebra

If you're familiar with the idea of diagonalization, then since $S$ satisfies $S^2=S$ and is an operator on $\mathbb{R}^2$, the characteristic polynomial of $S$ is then $p(z)=z^2-z$. Since $S\neq I$ ...
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1 vote

Simultaneously conjugate finitely matrices

Yes, if $F$ is infinite. The set of equations $MA_i=B_iM,i=1,...,k$ is a linear system $CX=0$, where $C$ has entries in $F$ and $X=\pmatrix{m_{11} \cr m_{12}\cr \vdots \cr m_{nn}}\in F^{n^2}$. Let $V=\...
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Simultaneously conjugate finitely matrices

Consider the monoid $G$ generated by $A_1, \dots, A_n$. We have two modules over the monoid algebra $F[G]$ just like in your solution to a special case. These modules are isomorphic after we base ...
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1 vote

Is following matrix diagonalizable?

There are two cases explicitly: If $b=c$, then $A$ is symmetric with real eigenvalues and hence always diagonalizable over $\mathbb C$ as well as $\mathbb R$. If $b\ne c$, then $A$ is non-symmetric. ...
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Matrix for reflection about the line $y = \tan (\theta) \, x$

The transformation\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix} maps any arbitrary vector $P=\begin{pmatrix}\alpha\\\beta\end{pmatrix}\in\mathbb R^2$...
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1 vote

How do I prove that a matrix is equals to 0?

As mentioned in the comments, you are being asked to show that the determinant of this matrix is zero. This can be shown if the rank of the matrix is $<3$. You can do this by showing the rows are ...
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-1 votes

How do I prove that a matrix is equals to 0?

Hint: If you add the top row to the bottom row, and then subtract two copies of the centre row from the bottom row, what do you get? And what is the determinant of a matrix of that type? On notation: ...
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Why is the $90$ degree clockwise rotation matrix not representative of the locations of $\hat\imath$ and $\hat\jmath$?

Let us label the matrices by $$R=\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \qquad R^{-1} =\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \newcommand{\i}{\hat\imath} \newcommand{\j}{\...
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How can I prove that induced norm of a row-stochastic matrix $A$, with respect to $\lVert \cdot\rVert_{\infty}$ is equal to one.

Each coordinate of $Ay$ is a convex combination of the coordinates of $y$. So, $\|Ay\|_\infty \leq \|y\|_\infty$ for each $y$. Then, choosing $y$ with all coordinates equal to some constant $\alpha$, ...
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Trivial question on derivative of quadratic form of vector-valued function

I found a related answer here: https://math.stackexchange.com/a/3128040/527323 Given a differentiable vector field $\mathrm f : \mathbb R^d \to \mathbb R^d$ and a matrix $\mathrm C \in \mathbb R^{d \...
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Hodge star operator : inner product of $k$-forms independent of orthogonal frames

$ \newcommand\Ext{{\textstyle\bigwedge}} \newcommand\trans[1]{#1^{\mathrm T}} \newcommand\form[1]{\langle#1\rangle} \newcommand\K{\mathbb K} \newcommand\Tensor{{\textstyle\bigotimes}} \newcommand\...
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Trace in a finite dimensional $C^*$-Algebra

You are missing a crucial part of the statement from the book, which is that $r$ is rational. If $r$ is irrational, then the embedding you are looking for does not exist; the reason is that a matrix ...
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1 vote

Solution of $XA=BX$?

you can re-write it as $A=X^{-1} B X$ , which is similiar to diagonalization of A, so you can solve one special kind of problem where $B$ is diagonal. $A=P^{-1} D P$ $=>$ $D$ is eigenvalues and $D$ ...
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What is the inverse of $A^{-1}+B^{-1}+C^{-1}$ for square matrices $A,B,C$?

Such a generalization does exist, and you're already 2/3rds of the way there with your case for two matrices. You've shown that if $A$, $B$, and $A+B$ are invertible, then so is $A^{-1} + B^{-1}$. Now ...
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How to find the linear codes of a linear generator matrix?

The example is over $\mathbb{Z}/3\mathbb{Z}$, so any modular reductions should be modulo 3!
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Proper way to represent matrix concatenation with indexed matrix

$ \def\o{{\tt1}} \def\bbR#1{{\mathbb R}^{#1}} \def\LR#1{\left(#1\right)} \def\size#1{\operatorname{size}\LR{#1}} \def\m#1{\left[\begin{array}{c}#1\end{array}\right]} $Let $$\eqalign{ m,n &= \size{...
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What do the (high) values in a controllability matrix mean?

Depending on the size of your system and the magnitude of the entries, the controllability matrix may be ill-conditioned and computing its rank will be prone to numerical errors. In other words, this ...
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Is there an easier formula for determinant and adjoint of matrix with this properties

To my knowledge: No, at least not by much, there are no hugely easier methods than the ones for general matrices. Obviously substituting $c=b$ in $\text{det}(A_{2\times 2})=ad-bc=ad-b^2$ simplifies a ...
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What happens with determinants in affine trasnformation?

A simplex in $n$ dimensions has $n+1$ vertices. To describe its shape and orientation but not its position, you can subtract the coordinates of one vertex from all the others. That's what's happening ...
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Matrices and Transformation

Check the geometric interpretation of the determinant and find the matrix of that tranformation. The rest follows
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Exercise $3$, Section $3.C$ - Linear Algebra Done Right

Here is rapidly how one can get those bases. Let $n=\dim V$ and $\def\rk{\operatorname{rk}}\def\im{\operatorname{im}}r=\rk T=\dim\im(T)$ (that's your $m$), and $p=\dim W$. Choose a basis of $\ker(T)$ ...
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Exercise $3$, Section $3.C$ - Linear Algebra Done Right

You've got the right general idea but your proof is overly complicated. Just observe that $Tv_1,\ldots,Tv_m$ are linearly independent since if $$\alpha_1Tv_1+\cdots+\alpha_m Tv_m=0$$ then $$T(\...
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Correlation between linearly transformed vectors

Let $\mathcal N(\mu, \sigma^2)$ be the distribution of each entry of $T$. You can easily prove by the law of large numbers that: $$\frac{1}{N} e_i^{\dagger}T^{\dagger} Te_j \underset {N \to \infty}\to ...
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Notation for the ith row and column of a matrix

Here is a notation based on already standard notation: $({\bf A}^{\intercal})_{i}$ and ${\bf A}_{j}$ for the $i$-th row and the $j$-th column, respectively, of the matrix ${\bf A}$.
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Correlation between linearly transformed vectors

$T$ is a simulation of N i.i.d N dimensional random variables, therefore $T^\dagger T$ is a simulation of scaled identity matrix $NI$. Therefore your conclusion holds. The reason of low R2 for $x^\...
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Hodge star operator : inner product of $k$-forms independent of orthogonal frames

Follow Nicholas Todoroff's hint (generalized Cauchy-Binet Formula), we have $$\sum_{I}(\sum_{J}a_J Q(I,J))(\sum_{J }b_J Q(I,J)))=\sum_{I }\sum_{J,K}a_Jb_KQ(I,J)Q(I,K)=\sum_{J,K}\sum_{I }a_Jb_KQ(I,J)Q(...
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Are the determinants of these matrices always negative under these conditions?

See my answer on MO under this link: https://mathoverflow.net/questions/426170#428268.
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How to compute $e^{At}$ with a $2\times2$ parametric matrix.

For general $A,$ the trick would be to diagonalize $A.$ This particular matrix is easier if you know the general formula for linear recurrences. $A$ has the property that $A\begin{pmatrix}x\\y\end{...
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5 votes

Does $ x^\dagger M x = \alpha \; x^\dagger x,$ imply $M x = \alpha x$?

I assume that $M$ is a square matrix. Suppose that $x$ is not a right eigen-vector of $M$ (and non zero), set $\alpha=\frac{x^\dagger M x}{x^\dagger x}$ to obtain a counter example.
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