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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Piecewise-linear (or otherwise monotonic) interpolation as a matrix problem

Background: I'm hoping to find (or write) an algorithm to piecewise linear-interpolate large sets of unevenly sampled functions (10s of thousands of arrays of a thousand or so $x$ and $y$ pairs, where ...
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2answers
42 views

Are the groups $G:=\{A \in M(n,\mathbb R) : A=A^t\}$ i.e. the group ( under addition ) of symmetric matrices and $O(n,\mathbb R)$ isomorphic?

Let $G:=\{A \in M(n,\mathbb R) : A=A^t\}$ i.e. the group ( under addition ) of symmetric matrices ; Are $G$ and $O(n,\mathbb R)$ isomorphic ?
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4answers
536 views

What is a short exact sequence?

I'll just quote my book here so you can see the definitions I have: Suppose that you are given a sequence of vector spaces $V_i$ and linear maps $\varphi_i: V_i\to V_{i+1}$ connecting them, as ...
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1answer
21 views

Proof of: a vector space spanned by $r$ vectors has dimension $\leq r$

I am confused about this proof of this statement in baby Rudin (Theorem 9.2 in third edition pp. 205). If a vector space $X$ is spanned by r vectors, then dimension($X$)$\leq r$ The proof goes ...
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13 views

Change of basis matrix from monomials to Legendre-polynomials

This is an exercise from an old exam. The dot product is defined as $\langle p, q\rangle= \int_{-1}^{1}p(x)q(x)dx$. I have to build the orthonormal Basis $Q_{2}$ based on $P_{2}$ with the help of ...
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1answer
20 views

Find a Subespace such that.. [closed]

How can I find a subespace $H\subseteq R^3$ such that $\langle(1,0,-1)\rangle \oplus H = R^3$ and $H \cap \langle (0,1,0) \rangle = \{0\}$ ? Thank you
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1answer
21 views

Matrix of non-degenerate product invertible?

Let $V$ be a finite dimensional complex vector space and $$\langle\cdot,\cdot\rangle:V\times V\to\Bbb C$$ a non-degenerate symmetric bilinear form. If $v_1,\ldots,v_k\in V$ are linearly independent ...
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39 views

On oblath's theorem [closed]

it is just my first encounter about this topic ,it is the topic that my prof gave to me in my undergrad studies.I found it interesting but there are still parts(like theorem) in this topic which make ...
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1answer
15 views

$nD$ rotation around a general $(n-2)$-dimensional subspace

According the Rodrigues' Rotation Formula $3D$ rotation matrix $\in$ $SO(3)$ corresponding to a rotation by an angle $\theta$ about a fixed axis specified by the unit vector $\hat{\omega}=(\omega_x,\...
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27 views

minimize quadratic form

In question Minimize Energy using Gauss-Seidel method with successive over- relaxation., when $$ E = \sum_i \|I_i - \mathbf N_i^T\mathbf L\|^2 + \lambda\sum_{i,j}\|\mathbf N_i - \mathbf N_j\|^2 = \...
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1answer
11 views

distance between two points with varying speeds

A Man walked from his house to office at 5kmph and got 20 minutes late. if he had travelled at 7.5kmph, he would have reached 12 minutes early. The distance from his house to office is? Here s1 = ...
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1answer
16 views

Proof that left singular vectors in SVD are orthogonal, and proof of low-rank approximation

I've been reading about SVDs and have a couple questions. First, let $A\in \mathbb{R}^{n\times d}$ be a matrix with SVD $U\Sigma V^T$. Let $\sigma_i$ denote the $i$'th singular value, with ...
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3answers
62 views

Find a set of vectors {u, v} in $R^4$ that spans the solution set of the equations

Find a set of vectors {u, v} in $\mathbb R^4$ that spans the solution set of the equations $x - y + 2z +3w = 0$ $4x + 2y - z + 3w = 0$ $ u =\begin{bmatrix}\\\\\\\end{bmatrix}, v =\begin{...
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1answer
43 views

Is there a theorem that says that a function between finite dimensional spaces has a matrix representation?

Just the line in the question. Is there an actual theorem that states this or is it something that people just know. If there is such a theorem I would like to be able to cite it, that's the purpose ...
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1answer
49 views

Eigenvector corresponding to eigenvalue $ 1 $ of a stochastic matrix

I am trying to justify fact $ 5 $ in this link which states that if $ A $ is a column stochastic matrix, then $ A $ has eigenvalue $ 1 $ and a unique eigenvector such that all entries are either ...
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1answer
31 views

First Order Difference Equations - Using Eigenvectors/Values

I was reading some notes and there was the following section: Start with a given vector $\vec{u}_0$. We can create a sequence of vectors in which each new vector is $A$ times the previous vector: $$\...
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1answer
60 views
+50

Rank and null space of a particular block matrix.

Let $D_1, D_2 \in \mathbb{R}^{N \times N}$ be diagonal matrices with diagonals that are linearly independent vectors. Let $A, B \in \mathbb{R}^{N \times N}$ be rank-deficient matries. Define $S = \...
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2answers
69 views

I'm looking for a matrix $M$ with $\det(M)=a^2+b^2+c^2+d^2$

In order to show that $(a^2+b^2+c^2+d^2)(A^2+B^2+C^2+D^2)= \alpha^2+\beta^2+\gamma^2+\delta^2$ with $a,b,c,d,A,B,C,D,\alpha,\beta,\gamma,\delta \in \mathbb Z$. I would like to find a matrix with ...
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0answers
56 views

I hope you resolve the question with surrounding solution method [closed]

That we know that: $$(i-\sqrt 3)^x-(i+\sqrt 3)^y=2^{xy}$$ Find the value of: $x+y$ .
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2answers
29 views

Describe all solutions of Ax = 0 (2)

Let $A = \begin{bmatrix}1&-5&-3&2\\4&-20&-12&8\end{bmatrix}$ Describe all solutions of $Ax = 0$ $x = x_2 \begin{bmatrix}\\\\\end{bmatrix} + x_3 \begin{bmatrix}\\\\\...
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2answers
23 views

Logic of Elementary Row Operations to Create Equivalent Systems

Can anyone explain why the 3rd operation applied on a system creates an equivalent system with the same solution. Elementary Row Operations. 1. Interchange two rows. 2. Multiply a row with a ...
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2answers
53 views

Describe all solutions of Ax = 0

Let $A = \begin{bmatrix}1&-5&3&-3&-4&-2\\0&0&1&1&0&-5\\0&0&0&0&1&-3\\0&0&0&0&0&0\end{bmatrix}$ Describe all ...
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2answers
24 views

Expressing the orthogonal projections on a linear operator $T$'s eigenspaces as polynomials in $T$

In the inner product space $\mathbb{C}^{2}$ with its standard inner product, let $$ T\begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} 3x+4y\\-4x+3y \end{pmatrix} $$ a linear operator. Express the ...
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1answer
21 views

Proof Relationship between System Solution and Matrix Rank [closed]

Can anyone prove this theorem? Suppose a system of m equations in n variables is consistent, and the rank of a augmented matrix is r. (1)The set of solutions involves exactly n-r parameters. (2)If ...
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2answers
22 views

The trace functional and its scalar multiples [duplicate]

I am trying to solve the following problem: Show that the trace functional on $n \times n$ matrices is unique in the following sense. If $W$ is the space of $n \times n$ matrices over the field $F$ ...
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1answer
24 views

Find a Jordan basis for the endomorphism $g:M_2(R)\longrightarrow M_2(R)$ such that…

Find a Jordan basis for the endomorphism $g:M_2(R)\longrightarrow M_2(R)$ such that $M(g,B) = \begin{pmatrix} 2&0&3&0\\ 1&2&0&3\\0&0&2&0\\ 0&0&1&2 \...
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0answers
17 views

stopping criteria for power-iteration to find rank-1 matrix

I start with B=I, A positive matrix, and compute B=(BA)/norm(B) by iterating until B is sufficiently close to rank-1 matrix. What is a good stopping criterion for this algorithm? There's Birkhoff ...
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3answers
27 views

Expressing a vector v as a linear combination of x and y

Express the vector $v = \begin{bmatrix}49\\0\end{bmatrix}$ as a linear combination of $x = \begin{bmatrix}6\\5\end{bmatrix}$ and $y = \begin{bmatrix}-5\\4\end{bmatrix}$ $v = $ ____ $x + $ ______$...
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3answers
38 views

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
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3answers
68 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 ...
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2answers
59 views

Algebra, linear transformation, minimal polynomial [closed]

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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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'...
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1answer
84 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 $T_n v = 0$. ...
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0answers
16 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 ...
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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 ...
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1answer
40 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 ...
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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
36 views

Help solving the equation [closed]

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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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 ...
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2answers
308 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-\...
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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
54 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 ...
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2answers
20 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
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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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38 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
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2answers
77 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*...
1
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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 (...