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 ...
3
votes
3answers
67 views

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 ...
0
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2answers
59 views

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 ...
0
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0answers
33 views

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'...
9
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1answer
83 views

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 $T_n v = 0$. ...
1
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0answers
14 views

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 ...
0
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2answers
48 views

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 ...
1
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1answer
38 views

Lipschitz continuity of $\sqrt{A}$

Let $U \subset\mathbb{R}^n$ be an open set, $\mathbb{S}^n$ be the set of all $n\times n$ symmetric real matrices, $A:U\to \mathbb{S}^n$ be a uniformly Lipschitz continuous function. Suppose $\exists ...
1
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0answers
19 views

If a vector subspace is in a union of other subspaces, then it's contained in one of them [duplicate]

Problem: Let $V$ be a finite dimensional vector space and $V_1,\ldots,V_n\subset V$ vector subspaces. Show that if $W\subset V$ is a vector subspace and $$W\subset V_1\cup\cdots\cup V_n,$$ then $...
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1answer
35 views

Help solving the equation [on hold]

I'm stuck and don't know what to do next to solve this equation. Any hints? $y(x_2−x_1)−y_1(x_2−x_1)=x(y_2−y_1)−x_1(y_2−y_1)$
0
votes
0answers
47 views

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 ...
0
votes
2answers
22 views

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 ...
5
votes
2answers
305 views

Algebraic multiplicity = geometric multiplicity?

I was wondering if algebraic multiplicity was equal to the geometric multiplicity. If the matrix (of size $n\times n$) is diagonalisable, i.e. the characteristic polynomial is of the form $$p(x)=(x-\...
0
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0answers
23 views

Relationship between geometric multiplicity, algebraic multiplicity and left and right eigenvectors of a matrix

The following statement is from the book Matrix Analysis by Horn and Johnson. An eigenvalue λ with geometric multiplicity 1 can have algebraic multiplicity 2 or more, but this can happen only if ...
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1answer
14 views

Help with some calculations

My question is: what I need to do to get 2nd equation from the first? 1) $TP1 = vp1 · λ + TS1$ $TP2 = vp2 · λ + TS2$ 2)$$TP_2 − TS_2 =\frac{vp2}{vp1}(TP1 − TS1)$$
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1answer
52 views

Matrix addition and eigen values/vectors

If I start with matrix A given by $A = \begin{bmatrix}a & b \\ c & d \end{bmatrix}$ and I express it as a sum $A = \begin{bmatrix} w & x \\ y & z ...
0
votes
2answers
19 views

Solving for x in a matrix equation?

I'm confused, how exactly can I solve this? I have no clue where to start Solve for $X$ $\begin{bmatrix}6&8&-6\\1&7&2\end{bmatrix} = 2X - 3\begin{bmatrix}-5&-2&-6\\4&...
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0answers
20 views

Find an orthogonal basis for the bilinear form

Find an orthogonal basis for the bilinear form over $\mathbb{R}$ given by $(x,y)\to x^tAy$ where $$A=\begin{pmatrix}1&4&4\\4&4&10\\4&10&16 \end{pmatrix}.$$
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0answers
37 views

Challenging calculation of a Jacobian for an unusual matrix coordinate transformation

I am studying a random matrix ensemble and I am having trouble performing a coordinate transformation. My question is very straightforward, but perhaps a bit technical. I have the following integral--...
2
votes
2answers
76 views

A question about orthogonality

Let $\mathcal{A}$ be a unital $*$-algebra over $\mathbb{C}$ and let $a,b\in\mathcal{A}$ be projections, that is, $a=a^*=a^2$ and $b=b^*=b^2$. If $a+b=1$, then $ab=0$. This follows from - \begin{align*...
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2answers
36 views

Algebraic or geometric multiplicity?

I am reading a proof of the fact that every linear transformation $L:V\to V$ can be represented by an upper triangular matrix $M$, with eigenvalues on the diagonal. And if the algebraic ...
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0answers
29 views

How long would it take in years to spend 50,000 and only spending 50 dollars a day? [closed]

A person has won 50,000 dollars and doesn't want to spend it in a short amount of time. Instead this individual has decided to spend 50 each day from the 50,000 he or she has. How long would it take (...
1
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1answer
35 views

Why this constant $c$ is the determinant of the operator

I'm reading Linear Algebra written by Kenneth M Hoffman and Ray Kunze and on page 172 he states the following corollary: Afterwards, he said that: Of course, the element $c$ in the last ...
3
votes
2answers
95 views

What kind of $n^{th}$ order polynomials are solvable by a square matrix with integer entries?

Consider a polynomial (monic for simplicity): $$x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n=0$$ Here we assume the roots are complex numbers. $a_k$ are integers. Now consider the corresponding matrix ...
0
votes
1answer
33 views

Reducing the Matrix to Reduced Row Echelon Form

Reduce the matrix $\begin{bmatrix}1&-1&-6\\4&-1&-15\\-2&2&12\end{bmatrix}$ to reduced row-echelon form How is my answer incorrect? I performed the row operations: 1) $R_2 =...
0
votes
1answer
15 views

Area of the region bounded by four vectors.

I'm stuck on how to approach this problem. I have a feeling it involves determinants and linear algebra. It's to find the area of the region bounded by the vectors: [-7,7], [5,5], [3, -4], [-5,-6]
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0answers
34 views

Linear equation [duplicate]

I have been troubles with the problem below, The line 4x-5y+20=0 cuts the x axis at A(0,4) and the y axis at B(-5,0). Find the equation of the median through O of triangle OAB. Find the equation of ...
1
vote
3answers
118 views

Prove that there are not two matrices 2x2 such that: $AB-BA=I_2$

I tried this question by multiplying explicitly the matrices but I think I'm not getting anything, so I think, well let's suppose false so $C(AB-BA)=C$ and find a contradiction but also I'm not ...
0
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1answer
35 views

Variant of Holder's inequality: $\|x\|_p \le n^{\frac1p- \frac1r} \|x\|_r$

So far I believed that only the reverse Holder inequality holds for $0<p<r<1,$ but then a student pointed out to me that $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ A few numerical ...
2
votes
1answer
45 views

Proof of Isometry and Reflection

$ V = \mathbb R^n$ is provided by the standard scalar product and by the standard basis $S$. $ W \subseteq V $ is a vector subspace and $ W^\bot$ is its orthogonal complement. a) Prove that there ...
1
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2answers
25 views

How to normalize and inverse a vector so it sums to 1 ?

I understand how normalization works. You sum up the individual values of the vector, you divide each value by the sum, and voila... they sum to 1. Why doesn't it work when you subtract them from ...
1
vote
1answer
48 views

How to solve proportions involving vector cross products?

I have the following proportion $\vec{JV} \times \vec{F_v} = \vec{JM} \times \vec{F_m}$ and all members are known except the magnitude of the vector $\vec F_m$, like described by another question here ...
3
votes
1answer
77 views

Tricky norm-inequality $\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$ for $p \in (0,1)$

For $r>p \ge 1$ one can show that in $\mathbb{C}^n$ we have $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ My question is now: Does this also hold for $1 \ge r>p>0$? Obviously we ...
0
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2answers
37 views

Solving the system using elimination

Solve the system using elimination $-6x -2y + 3z = 34$ $-5x -4y + 4z = 32$ $2x +5y -4z = -19$ $x = ?, y = ? , z = ?$ So I threw this in a augmented matrix and put it in REF form ...
0
votes
3answers
62 views

Show algebraically that the graph of $y=x^2 + kx - 2$ will cut the $x$-axis twice for all values of $k$

A quadratics question. Show algebraically that the graph of $y=x^2 + kx - 2$ will cut the $x$-axis twice for all values of $k.$ I recently asked a similar question, but this problem seems ...
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votes
1answer
42 views

Finding the equation of a median through a triangle. [closed]

The line $4x-5y+20=0$ cuts the $y$ axis at $A(0,4)$ and the $x$ axis at $B(-5,0)$. Find the equation of the median through O of triangle $OAB$. Note, the book indicates the answer as $4x-5y=0$ while ...
0
votes
1answer
26 views

Having trouble understand Row Echelon Form

I'm having trouble understanding Row Echelon Form. I'm trying to solve the system $-2x - 10y - 29z = 5$ $-4x - 19y -56z = -3$ $x + 5y + 15z = 3$ it has the solution $x = ? , y = ? , ...
1
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1answer
36 views

Why solving linear equations is taking a quotient by some subspace?

Linear equation can be represented by a linear form, and its solution space is the same thing as kernel of this form. The same is true for system of linear equations. But this lecture notes suggest ...
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0answers
24 views

Module over nonassociative algebra [on hold]

I have $A$ an algebra $*$ that is commutative an have an identity $(a^2b)b=((ab)ba$ and asking me verify conditions conditions for a $A$-module $*$ $M$. How I would use Linearization of identity? $...
0
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1answer
21 views

Confused with the reexpression of a Hamiltonian in eigenbasis

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring of spin chain. To compute the complexity of ...
0
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1answer
47 views

Matrix with all entries N

Is there a specific name for a matrix where all entries are the name number? I am writing a program where I want to be able to describe a matrix like this in the same way I would the identity matrix, ...
2
votes
3answers
53 views

Define $L(A) = A^T,$ for $A \in M_n(\mathbb{C}).$ Prove $L$ is diagonalizable and find eigenvalues

Let $L:M_n(\mathbb{C}) \to M_n(\mathbb{C})$ be defined by $L(A) = A^T,$ where $A^T$ is the transpose of $A$ and $M_n(\mathbb{C})$ is the space of all $n \times n$ matrices with complex entries. Prove ...
10
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3answers
100 views

Can one factor matrices?

I know that one can factor integers as a product of prime numbers. Is there an analog of it to matrices? Can we define prime matrices such that every matrix is a product of prime matrices? Is there ...
1
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2answers
69 views

How to determine which of the following matrices are similar?

If we have the following three matrices: $$ A=\begin{bmatrix} 7 &1 \\ -5 &3 \end{bmatrix},\;\; B=\begin{bmatrix} 5 &-1 \\ 1 &5 \end{bmatrix},\;\; C=\begin{bmatrix} 5 &1 \\ 1 &...
2
votes
3answers
123 views

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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1answer
60 views

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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0answers
56 views

$2 \times 2$ 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=\...
0
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1answer
31 views

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 ...
1
vote
1answer
32 views

$V = W_1\oplus W_2\oplus… \oplus W_s$ 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,...
0
votes
1answer
48 views

Bilinear form vs two-form

I have a basic linear algebra question since I'm confused with the definitions: What is the difference between bilinear form and 2-form? I looked up in Wikipedia and it says that a linear form $B(v,...