# Tagged Questions

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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### Finding a and b in a matrix

Find $a$ and $b$ such that $\begin{bmatrix}-11\\9\\-12\end{bmatrix} = a \begin{bmatrix}1\\-3\\3\end{bmatrix} + b \begin{bmatrix}7\\3\\0\end{bmatrix}$ I think it's trivial that $a = -4$, which is ...
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### Prove that for any diagonalizable matrix $A$, $A^n$ is diagonalizable and also $aA^m+bA^n$

Suppose that A is a diagonalizable matrix. 1) Prove that $A^n$ is diagonalizable 2) Prove that $aA^n + b A^m$ is diagnalizable, for every $a,b\in\mathbb{K}$ I thank you any help or hint you can ...
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### Algebra, linear transformation, minimal polynomial [on hold]

Let $T : M_{n×n}(\Bbb F) \to M_{n×n}(\Bbb F)$ the linear transformation defined by $T (A) = AB$, for some matrix $B \in M_{n×n}(\Bbb F)$ fixed. Show that the minimal polynomial of $T$ coincides with ...
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### Given $A = \Sigma\lambda uu^H$. $A = -A^H$. Prove $\lambda$ is imaginary

Given $A = \Sigma\lambda uu^H$. and $A = -A^H$. Prove $\lambda$ is pure imaginary. (Btw, $u$ are orthonormal vector, don't know how to write here in math-stackexchange with the ^) I've two proofs I'...
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### Does $\forall v ( T_1 v = 0 \lor T_2 v = 0 \lor \dots \lor T_n v =0 )$ imply $T_1 = 0 \lor T_2 = 0 \lor \dots \lor T_n = 0$?

Let $V$ and $W$ be vector spaces and $T_1$, $T_2$, $\dots$, $T_n$ be linear transformations from $V$ to $W$, such that for every $v$ in $V$, either $T_1 v = 0$, $T_2 v = 0$, $\dots$ or $T_n v = 0$. ...
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### Auslander-Reiten theory: exercise $23.b$ of 'Elements of the Representation Theory of Associative Algebras'

I am solving exercise $23.b$ of chapter IV of 'Elements of the representation theory of associative algebras' by Assem, Simson and Skowronski. The question is the following: Consider the following ...
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### How to describe range of a linear transformation?

I'm self studying Linear Algebra from Hoffman Kunze, and I've come upon this problem. With complex number $z=x+iy$, $$T(z)=\begin{pmatrix} x-7y & 5y \\ -10y & x+7y \\ \end{pmatrix}$$ is ...
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### Help solving the equation [on hold]

I'm stuck and don't know what to do next to solve this equation. Any hints? $y(x_2−x_1)−y_1(x_2−x_1)=x(y_2−y_1)−x_1(y_2−y_1)$
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### Find the eigenvalues the block matrix $M=\begin{bmatrix}A+2D & A \\ A & D \end{bmatrix}$

Let $A$ be any square matrix with eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n$ and $D$ is a diagonal matrix with entries $d_1,d_2,\cdots,d_n$, then how can one find the eigenvalues of the ...
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### Which one is equation of tangent

Is equation of tangent plane $z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0} )$ or $z=f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0} )$ In my book I found ...
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### Importance of the homogeneity assumption in definition of linear map

Let $V$ and $W$ be vector spaces over field $F$. A function $f: V \rightarrow W$ is said to be linear if for any two vectors $x$ and $y$ in $V$ and any scalar $\alpha\in F$, the following two ...
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### Calculating inverse function with 2 variables

$f: R^{2}\mapsto R^{2}$ $(x,y)\mapsto (x^{2}-4y^{2}+x, -xy+3y)$ I should calculate inverse function of $f$ in point $(3,1)$. I tried to do $(x,y)\mapsto(u,v)$, but I just dont know how to get x ...
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### $2 \times 2$ block matrix related

let $A$ be any matrix of order $n$, $J$ is matrix of order $n$ whose all entries are $1$, and $I$ is an identity matrix of order $n$, then how to find eigenvalues of following block matrix? M=\...
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### For two square positive-semidefinite matrices $A$ and $B$, does the relation $|A|^{-\frac{1}{2}}|B|^{-\frac{1}{2}} = |AB|^{-\frac{1}{2}}$ hold?

For two square positive-semidefinite matrices $A$ and $B$, does the relation $|A|^{-\frac{1}{2}}|B|^{-\frac{1}{2}} = |AB|^{-\frac{1}{2}}$ hold? Here the absolute value signs are the determinant ...
### $V = W_1\oplus W_2\oplus… \oplus W_s$ we must have $W = (W_1 \cap W )\oplus (W_2\cap W)\oplus … (W_s \cap W)$?
Let $T$ be a linear operator on an $n-$dimensional vector space $V$ ($n > 1$) and $W$ is a $k-$dimensional $(0 < k < n)$, $T-$invariant subspace. Show that if $T$ has $n$ distinct eigenvalues,...