Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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4
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2answers
54 views

is this subset a subspace - redux

OK, I have been bothering people here with this for days and with luck I finally have this. People have helped a lot here so far. (Doing these examples is I hope helping me learn the proofs, but I ...
2
votes
5answers
136 views

Finding the limit of a matrix

Suppose that $A=\begin{pmatrix}4&1 & 5\\ 2& 7& 1\\ 2& 2& 6\end{pmatrix}.\;$ How can I find $\;\displaystyle \lim_{n\to\infty}A^n$? What theorem(s) should I use to solve this? ...
4
votes
2answers
477 views

An operator that commutes with another operator $T$ with distinct characteristic values is a polynomial in $T$

I'm trying to solve some problems in Hoffman and Kunze and I'm kind of stuck on this one. This is 6.5.3 on Hoffman and Kunze. Here is the question: Let $T$ be a a linear operator on an ...
9
votes
5answers
629 views

How to prove that the set $\{\sin(x),\sin(2x),…,\sin(mx)\}$ is linearly independent?

Could you help me to show that the functions $\sin(x),\sin(2x),...,\sin(mx)\in V$ are linearly independent, where $V$ is the space of real functions? Thanks.
1
vote
3answers
108 views

Suppose that a $3\times 3$ matrix $M$ has an eigenspace of dimension $3$. Prove that $M$ is a diagonal matrix.

How would I go about this? I realise that having dimension 3 means that the solution to $(A-\lambda I)\mathbf b = \mathbf 0$ has 3 free parameters, which would in turn mean that $(A-\lambda I)$ is the ...
3
votes
2answers
149 views

if $A$ is Abelian group , $B$ is subgroup of $A$ , Is $B \times A/B \cong A$? [duplicate]

If $A$ is abelian group and $B$ is a subgroup of $A$, $B$ is normal subgroup of $A$. Is it true that $B \times A/B \cong A$? I ask because I was watching an online lecture from a course in abstract ...
6
votes
0answers
148 views

Dimension of the space of algebraic Riemann curvature tensors

Given $n\in \mathbb N$, consider the vector space $\mathbb R^{n^4}$ whose elements I will denote by $(R_{abcd})$ with indices $a,b,c,d \in \{1, \dots, n\}$. This vector space is $n^4$-dimensional. The ...
2
votes
1answer
33 views

Relationship between $(L|_M)^*:N^*\to M^*$ and $L^*|_{N^0}:N^0\to M^0$?

Suppose $L:V\to W$ is a linear transformation, and $L(M)\subseteq N$ for some subspaces $M\subseteq V$ and $N\subseteq W$. A question I'm reading asks rather open-endedly if there is a relationship ...
3
votes
2answers
528 views

Given a linear transformation matrix, T, find the equation for the curve that T transforms a circle into.

Given the linear transformation matrix: $$T=\pmatrix{2&-3\\1&1}$$ Find the equation for the curve that $T$ transforms a circle with equation $x^2+y^2=6$ into. What I know: My basis is going ...
1
vote
2answers
55 views

Derivative of column-row multiplication

How can I take derivative $$\frac{d}{dA}(x - Ab)(x - Ab)^T$$ where $x$ and $b$ are known vectors of the same size and matrix $A$ is symmetric and positive-definite? Update: This expression could be ...
2
votes
0answers
204 views

Proof of the spectral theorem for normal operators from two lemmas

I have the following lemmas that I can prove: Let $T$ be a linear operator on a Hermitian space $V$ and let $W$ be a $T$-invariant subspace of $V$ . Then $W^⊥$ is $T^*$-invariant Let $T$ be a normal ...
1
vote
0answers
118 views

The vector space of polynomials

I was given a theorem: The polynomials (where $f$ and $g$ are complex polynomials of degrees $n$ and $m$) $$f(z), zf(z), \ldots , z^{m−1}f(z), g(z), zg(z), \ldots,z^{n−1}g(z)\tag{7.6.4}$$ ...
0
votes
1answer
111 views

How to show all eigenvalues are positive?

Could you help me to show that the following matrix has all its eigenvalues positive? $$H= \begin{bmatrix} \sum_{k=1}^ng_1(x_k)^2 & \sum_{k=1}^ng_1(x_k)g_2(x_k) & \cdots & ...
-1
votes
1answer
58 views

Proving that a matrix is diagonalizable

Let $ T $ be the linear operator on $ \Bbb R^3 $ which is represented by the matrix $$ A = \begin{bmatrix} 6 & -3 & -2 \\ 4 & -1 & -2 \\ 10 & -5 ...
0
votes
1answer
93 views

Determine all functions for a given gradient

I don't really have an approach to solve this problem so it would be very kind if you could tell me what to do first: Determine all functions $f:\mathbb{R}^2 \to \mathbb{R}$ for which applies: ...
0
votes
1answer
43 views

eigenvalue and independence

Let $B$ be a $5\times 5$ real matrix and assume: $B$ has eigenvalues 2 and 3 with corresponding eigenvectors $p_1$ and $p_3$, respectively. $B$ has generalized eigenvectors $p_2,p_4$ and $p_5$ ...
0
votes
2answers
198 views

Linear algebra T- invariant subspaces

Let $ T $ be the linear operator on $ \Bbb R^3 $ which is represented by the matrix $ A = $ $ \begin{bmatrix} 2 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & ...
1
vote
1answer
61 views

Help with anti-image matrix

First of all, I am very sorry but I don't know the mathematics terminology in English, so I'll try to explain as good as i can but i will probably do some mistakes since it's not my native language. ...
3
votes
0answers
32 views

Existence of weights of a finite dimensional representation of a semisimple Lie algebra

Let $\mathfrak{g}$ be a semisimple complex Lie algebra. I want to show that every finite dimensional irreducible representation of $\mathfrak{g}$ is a weight module, and I need the existence of at ...
1
vote
2answers
144 views

How to show that a given set is a subspace

OK I just want to be sure I have done this correctly. Given: $R^3$, are the following sets subspaces? (a) $W_1$ = {($a_1$,$a_2$,$a_3$) $\in R^3: a_1 = 3a_2$ and $a_3 = -a_2$ Since the set you get ...
1
vote
2answers
101 views

Show that this set is linearly independent

Ñotation: $V$ is a vector spaces of real functions $g:X\rightarrow\mathbb{R}$; $\{g_1,...,g_m\}$ is a subset of $V$; $\{x_1,...,x_n\}$ is a subset of $X$, where $x_i\neq x_j$ when $i\neq j$; ...
1
vote
3answers
61 views

Symmetric Matrices Help

I'm having a bit of trouble with the following question. Suppose $A$ is a square matrix. a) Show that the matrix $B = A+A^T$ is symmetric. Not sure how to do this. But here is my attempt. Well, let ...
1
vote
1answer
64 views

Orthonormal basis, isomorphism preserving dot product

For an orthonormal basis $v_1, ..., v_n$ of $(V, \cdot )$ $\mathbb{R}^n \ni (x_1, ..., x_n) \rightarrow \sum x_jv_j \in V$ is an isomorphism preserving dot product. I've already proven that it ...
2
votes
3answers
529 views

Are the eigenvalues of $A^\top A$ equal to those of $AA^\top$?

In an exam question I was asked to calculate the eigenvalues of $A^\top A$, where $A = (a_1\ a_2\ a_3); a_1=(0\ 2\ 1)^\top; a_2=(1\ -1\ 1)^\top; a_3=(1\ 1\ -1)^\top;$ and $A^\top$ stands for the ...
0
votes
1answer
131 views

Jacobian determinant and orientation

So in Jacobian determinant, it is often said that it gives information about whether Jacobian matrix changes orientation, but I cannot get what orientation exactly in this context.
2
votes
0answers
68 views

Ideals (one-sided ideals) of $n×n$ upper triangular matrices

Is there any characterization of ideals (one-sided ideals) of $n\times n$ upper triangular matrices? I have just seen in monthly journal about $2 \times 2$ matrices in the below article Left and Right ...
0
votes
0answers
77 views

Finding if the Point is in the triangle

Can You Please tell me if this is right Point(x,y,z) triangle points ABC using co-planer determinant |x-Ax y-Ay z-Az| |x-Bx y-By z-Bz| = 0 |x-Cx y-Cy z-Cz| then the point is in the tringle
1
vote
3answers
122 views

How to convert a permutation group into linear transformation matrix?

is there any example about apply isomorphism to permutation group and how to convert linear transformation matrix to permutation group and convert back to linear transformation matrix
0
votes
1answer
1k views

Projection matrix onto null space

I have a matrix H and I want to find the projection matrix onto null space. How can I do this? Sorry if my question seems naive. Thank you, Tanja
1
vote
1answer
50 views

Proof convex polyhedron with line does not contain a corner if closed

The excercise I am struggling with is the following: Given a convex closed polyhedron that contains a line, the question is, whether this polyhedron can also contain a corner. My idea was to make a ...
1
vote
1answer
96 views

Matrix Diagonalization - Eigenvectors

I'm not sure that I've understand what I'm doing when I'm trying to diagnolize the matrice.Maybe one of the reason of this is, I can't think it geometrically or I can't understand the purpose of this. ...
1
vote
1answer
100 views

Existence of a common minimizer

Do \begin{equation} \sum_{i=1}^{3}\left|\lambda_{i}\left(\operatorname{diag}\left(1,3,5\right)-U\operatorname{diag}(2,4,6)\,U^{T}\right)\right| \end{equation} and \begin{equation} ...
0
votes
3answers
232 views

Finding a unique representation as a linear combination

ok, another problem suggested by my prof. the vectors $u_1 = (1,1,1,1)$, $u_2 = (0,1,1,1)$, $u_3 = (0,0,1,1)$, $u_4 = (0,0,0,1)$, are a basis for $F^4$. Find a unique representation of an arbitrary ...
5
votes
2answers
418 views

$A$ is some fixed matrix. Let $U(B)=AB-BA$. If $A$ is diagonalizable then so is $U$?

This is from Hoffman and Kunze 6.4.13. I am studying for an exam and trying to solve some problems in Hoffman and Kunze. Here is the question. Let $V$ be the space of $n\times n$ matrices over a ...
0
votes
1answer
99 views

Could someone help me to prove that this symmetric matrix is definite positive?

Let $a_{ij}\in\mathbb{R}$ for all $i,j\in\{1,...,n\}$ and $m\in\mathbb{N}$. Consider the matrix below. $$B=\begin{bmatrix} \sum_{k=1}^n(a_{1k})^2 & \sum_{k=1}^na_{1k}a_{2k} & \cdots & ...
2
votes
2answers
5k views

How to check if a set of vectors is a basis

OK, I am having a real problem with this and I am desperate. I have a set of vectors. {(1,0.-1), (2,5,1), (0,-4,3)} How do I check is this is a basis for $R^3?$ My text says a basis B for a vector ...
2
votes
0answers
132 views

This matrix is an attractor?

I'm trying to find for which values of $\gamma$ the matrix A is an attractor: $$ A=\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ -1 & 0 & \gamma ...
1
vote
0answers
33 views

the det of identity replacing column

Let $A\in{\mathbb{R}}^{n\times n}$, $b\in\mathbb{R}^{n}$, and $x\in\mathbb{R}^{n}$ be given, where $A$ is nonsingular and $Ax=b$ holds. Let $X_{j}$ be the matrix obtained from the $n\times n$ identity ...
4
votes
1answer
91 views

$\delta$ Notation in linear algebra

In this equation below, what is $\delta_{l,q}$ denoting? Is $\delta$ a standard notation, or anything to do with all one's or the basis matrix etc? $$A_{ij}=\delta_{l,q}\left(\sum_{h=1}^n B_{l,h} + ...
5
votes
4answers
269 views

Every element of $U + V + W$ can be expressed uniquely in the form $u + v + w$

Suppose that $U$, $V$ and $W$ are subspaces of some given vector space. With the obvious definition of $U + V + W$, show that every element of $U + V + W$ can be expressed uniquely in the form $u + v ...
1
vote
1answer
25 views

Given a specefic set $ A$ we need to find $A^\perp$

Suppose we have a set of functions which are an element of $L^2[0,1]$ where if we let f(x) be the function equal to 0 from $0<x<1/2$. If this set A is a subset of the Hilbert space $L^2[0,1]$ ...
2
votes
0answers
88 views

Interpret the Parallelogram Law

I'm starting to teach myself linear algebra from a book but I don't understand the following question. Interpret $||A+B||^2+||A-B||^2 = 2||A||^2 +2||B||^2$ as a "parallelogram law." Any suggestions ...
5
votes
0answers
102 views

Is this a spectral decomposition/embedding/isometry?

Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$. Now if I take the same ...
1
vote
2answers
163 views

Orthonormalize the set of functions {1,x}

I'm having trouble while doing this exercise, it says: In the vector space of the continuous functions in [0,1] with the inner product : $$\langle f,g \rangle = \int_{0}^{1}f(x)g(x)dx$$ a) ...
2
votes
0answers
108 views

Intersection of orthonormal basis of two subspaces

I want to ask how can I form a subspace which is an intersection of the orthonormal basis of two another subspaces and find its dimension in Matlab? Thank you in advance, Maya
2
votes
2answers
123 views

Finding eigenvectors of similar matrices

Suppose I have two matrices $B$ and $ABA^{-1}$ where $A = [v_{1}\, v_{2}\, v_{3}]$ with $v_{i}$ column vectors such that $\{v_{1}, v_{2}, v_{3}\}$ form an orthonormal basis of $\mathbb{R}^{3}$ (not ...
1
vote
0answers
42 views

Edge in a convex polytope

I want to show that a convex polytope $A$ that is an intersection of half-spaces contains an edge if $ A=\{x \in \mathbb{R}^n|Ax=0 \wedge x \ge 0\}$, where x greater equal 0 means, that all components ...
2
votes
1answer
238 views

Can we 'form' infinitely many subspaces out of finite dimensional vector space?

Let $V$ be a vector space over $\mathbb{R}$ of dimension $n$, and let $U$ be a subspace of dimension $m$, where $m < n$. Show that if $m = n − 1$ then there are only two subspaces of $V$ that ...
2
votes
2answers
45 views

How to calculate these matrices? - explanation of the procedure

Can you please help me solve this problem? I have got these matrices $A=\left(\begin{array}{ccc} 2 & 1 & 0 \\ 0 & 1 & 2 \\ 3 & 0 & 1 \end{array}\right) $, ...
0
votes
1answer
98 views

Symmetric positive definite with respect to an inner product

Let $A$ be a SPD(symmetric positive-definite) real $n\times n$ matrix. let $B=LL^T$ be also SPD. Let $(,)_B$ be an inner product given by $(x,y)_B=x \cdot By=y^T Bx$. Then ...