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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1answer
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Does a contractible set have contractible preimage, under a linear map?

Let $T:V\to W$ be a linear map of vector spaces, and let $A\subset W$ be contractible. Then is $T^{-1}(A)$ also contractible?
2
votes
1answer
30 views

Eigenvalue perturbation theory for $(A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$

Let $A, B$ be $n \times n$ matrices with full rank. I'm interested in getting a bound on how the smallest eigenvalue of $S = (A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$ changes when I perturb $A$ and $B$. ...
0
votes
1answer
21 views

Proving Polynomial is a subspace of a vector space

Picture of question. I'm really stuck on proving this question. I know that the first axioms stating that 0 must be an element of W is held, however I'm not sure how to prove closure under addition ...
0
votes
1answer
22 views

What does it mean for an inner product to be conjugate linear in the second entry?

Let $G$ be a group and $L^2(G) = \{f: G \rightarrow \mathbb{C} \}$. Now define an inner product on $L^2(G)$ by $$\langle f, g \rangle = \sum_{x \in G}f(x)\overline{g(x)}$$ Where $\overline{g(x)}$ is ...
1
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0answers
39 views

Frobenius-Perron dimension on a fusion category

Let $C$ be a fusion category with simple objects $V_i\in I$. The fusion rule is $V_i\otimes V_j \cong N_{i,j}^k V_k$. The Frobenius-Perron dimension of a simple object $V_i$, $\mathrm{FPdim}(i)$, is ...
2
votes
1answer
37 views

Transposition not diagonalizable in characteristic 2

In another thread it was proved that transposition as a linear map is diagonalizable. This, however, does not hold when we are working over a field of characteristic 2. I suppose the proof of this can ...
2
votes
1answer
41 views

What the gray area in the plane $x + y + z = 1$ means?

I'm studying Solving Systems of Linear Equations (Fraleigh - Linear Algebra), and I'm wondering for the meaning of the grayed are which appears in the following image: The author adds: We know ...
1
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1answer
29 views

Matrix equation implies invertibility

Let $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$ be a diagonal matrix with positive entries $\lambda_i > 0$ (some of them might coincide). If we have the matrix equation $A D A^t = ...
0
votes
1answer
36 views

Uniqueness of a linear operator

The wikipedia entry for bounded operators shows that for the space $X$ of trigonometric polynomials on $[-\pi,\pi]$ with norm $$\lVert P\rVert = \int_{-\pi}^{\pi}\lvert P(x)\rvert ...
-1
votes
1answer
39 views

isomorphic linear spaces [on hold]

Let $S$ be the space of $3\times 3$ skew-symmetric real matrices. Then $dim\ S=3$. Is it true that $S$ is a vector isomorphic to $\mathbb R^3$? What is the isomorphism then?
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votes
0answers
15 views

What method to use to find only positive solutions of an under-determined system of non-homogeneous linear equations? [on hold]

I am trying to solve an under-determined system of two non-homogeneous linear equations but I want to find only solution of the variables >0.
0
votes
3answers
35 views

Suppose $U$ and $W$ are subspaces of the vector space $V$. Show that $U + W$ = $sp(U \cup W)$

I am not sure where to start with this one, any help would be appreciated. I tried using the definition of a span but I couldn't see where to go from there.
2
votes
1answer
63 views

Possible to do well in Algebra without loving Analysis much? [on hold]

Having taken some courses in higher algebra, I realized that what I truly appreciate in mathematics is abstract algebra. But it also appears that I'm not a big fan of real analysis [at least I don't ...
1
vote
2answers
48 views

If $V$ is a vector subspace of $R^n$, prove that $V^ \bot$ is a vector subspace of $R^n$

Let $V$ be a subspace of $R^n$. Let $V^ \bot$ be a subset of $R^n$ defined by: $V ^ \bot$ = {$\vec x \in R^n$: $\vec x * \vec v = 0$ for all $\vec v \in V$} Prove that $V^ \bot$ is a subspace of ...
1
vote
1answer
26 views

How to Prove $V\otimes sl(k)=sl(V)$?

Let V be a vector space over a field $k$. Let $sl(n)$ be the set of all matrices elements from $k$ with trace zero. Is it true that $V\otimes _k sl(n)=sl(V)= \text{set of all $n\times$ n matrices ...
1
vote
1answer
48 views

Why the identity $P_X=P_XZ(Z'P_XZ)^{-1}Z'P_X$ with $P_X=X(X'X)^{-1}X'$?

Suppose $X$ and $Z$ are matrices such that $(X,Z)$ and $P_XZ$ both have full column ranks. Here, $P_X=X(X'X)^{-1}X'$. Consider a regression model $$ P_Xy=P_XZ\zeta+v\tag{A} $$ where OLS is used ...
1
vote
0answers
23 views

Extending the trace inner product to all matrix (real) inner products

In ${\bf R}^{n\times p}$ we have the trace inner product given by $$\langle A, B\rangle=\text{tr}(A^TB)$$ which can be interpreted as the Euclidean inner product on ${\bf R}^{np}$. All inner ...
3
votes
1answer
60 views

Trace of the $k$-th Exterior Power of a Linear Operator

Let $V$ be an $n$ dimensional vector space over a field $F$ and $T$ be a linear operator over $V$. Assume that the characteristic of $F$ is not $2$. Definition. Consider the map $f_1:V^n\to ...
92
votes
5answers
4k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
5
votes
1answer
65 views

What is so special about the Schwarz Inequality?

I am studying Spivak's Calculus and the first two problem sets have rather lengthy,but very interesting, work-throughs of three proofs for the Schwarz Inequality: $$\sum_{i=1}^{n} x_iy_i ...
7
votes
1answer
157 views

Proving positive definiteness of matrix $a_{ij}=\frac{2x_ix_j}{x_i + x_j}$

I'm trying to prove that the matrix with entries $\left\{\frac{2x_ix_j}{x_i + x_j}\right\}_{ij}$ is positive definite for all n, where n is the number of rows/columns. I was able to prove it for the ...
5
votes
1answer
50 views

Cayley-Hamilton Theorem - Trace of Exterior Power Form

Let $V$ be an $n$-dimensional vector space over a field $F$ (the characteristic of which, for the purpose of this post, may be taken as $0$). Let $T$ be a linear operator on $V$ and $\lambda\in F$. ...
3
votes
2answers
76 views

Determinant from Paul Garret's Definition of the Characteristic Polynomial.

$\DeclareMathOperator{\id}{id} \DeclareMathOperator{\End}{End}$ On pg. 390 of Paul Garret's notes on Algebra, a definition for the characteristic polynomial is given, which I discuss here. Let $V$ be ...
10
votes
2answers
217 views

A tricky problem about matrices with no $\{-1,0,1\}$ vector in their kernel

A Hankel matrix is a square matrix in which each ascending skew-diagonal from left to right is constant. Let us call a matrix partial Hankel if it is the first $m<n$ rows of some $n$ by $n$ Hankel ...
0
votes
1answer
36 views

2nd Order Differential Equation, particular integral query.

What would the form of the particular integral be of the following differential equation: $$\frac{d^2y}{dx^2} -4 \frac{dy}{dx} +5y=8 \sin x$$ Should the particular integral be of the form; ...
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votes
0answers
41 views

Dimension of the intersection of hyperplanes in a finite dimensional space [on hold]

Let the dimension of the vector space $V$ be $n$. Then how to prove that the dimension of the intersection of $k$ hyperplanes with $k<n$ is $n-k$.
5
votes
3answers
329 views

Calculate complex determinant

$$\left| {\begin{array}{*{20}{c}}{{a^2}}&{{{(a + 1)}^2}}&{{{(a + 2)}^2}}&{{{(a + 3)}^2}}\\{{b^2}}&{{{(b + 1)}^2}}&{{{(b + 2)}^2}}&{{{(b + 3)}^2}}\\{{c^2}}&{{{(c + ...
8
votes
2answers
401 views

This theorem about matrices of linear maps doesn't look correct.

Consider the following theorem: Theorem. Let $f\colon L\to M$ be a linear mapping of finite-dimensional vector spaces. Then there exist bases in $L$ and $M$ and a natural number $r$ such that the ...
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2answers
42 views

Plane Equation in R^3 from Column Space

Dear fellow mathematics enthusiasts, I require assistance on an easy excercise. I will try to be as brief and concise as possible. Problem description Let matrix $\mathbf{A}$ = $ \begin{bmatrix} 1 ...
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votes
2answers
17 views

Show a linear transformation on an orthogonal complement is a subset of the orthogonal complement. [on hold]

Let $V$ be a finite-dimensional complex inner-product space and let $W$ be a subspace of $V$. Also let $W^{\perp}$ be the orthogonal complement of $W$. Prove that if $T$ is a self-adjoint linear ...
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votes
0answers
31 views

Let P4 be the vector space of polynomials of degree at most 4. For the following map decide if it is an isomorphism?

How can i describe this operation as an isomorphism or not ? $p(x)=p_0+(p_1)x+(p_2)x^2+(p_3)x^3+(p_4)x^4\longmapsto q(x)=p_1+(p_2)x+(p_3)x^2+(p_4)x^3+(p_0)x^4$
0
votes
0answers
24 views

Writing a Rotation Matrix About an Angle

I am asked to find a rotation matrix $R_O$ of an angle $O$ about axis $u\in R^3$, with $u$ having length of 1. I've looked up this concept on the web but I have no idea where to get started...could ...
0
votes
2answers
45 views

Let $p(t), q(t) ∈ \mathbb C[t]$ be relatively prime, $A ∈ M_n(\mathbb{C})$. Show that $\operatorname{rank}(p(A))+\operatorname{rank}(q(A)) ≥ n$.

Let $p(t), q(t) ∈ \mathbb C[t]$ be relatively prime, $A ∈ M_n(\mathbb{C})$. Show that $\operatorname{rank}(p(A))+\operatorname{rank}(q(A)) ≥ n$. I have been stumped on this question for quite awhile. ...
0
votes
1answer
58 views

Let $V$ be a finite-dimensional vector space over a field $F$, and $f ∈ L(V )$. Show that there is an invertible $g ∈ L(V )$ such that $gfgf = gf$ [closed]

Let $V$ be a finite-dimensional vector space over a field $F$, and $f \in L(V )$. Show that there is an invertible $g \in L(V )$ such that $gfgf = gf$. Not sure where to start.. would someone ...
1
vote
1answer
29 views

Bounds for the eigenvalues of a matrix in a finite differences scheme

While implementing a numerical solution to a PDE with finite differences, the following scheme arises: $$v_{j+1} = Av_j$$ Where $$A =\begin{bmatrix} 1-4\lambda&(2+\mu ...
144
votes
6answers
10k views

Why does this matrix give the derivative of a function?

I happened to stumble upon the following matrix: $$ A = \begin{bmatrix} a & 1 \\ 0 & a \end{bmatrix} $$ And after trying a bunch of different examples, I noticed the ...
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vote
1answer
100 views

Is $\det (A + B)=\det (A) + \det (B) + \operatorname{tr}(A \operatorname{adj}(B))$?

Let $A,B \in {M_n}$. Is this true that $\det (A + B) = \det (A) + \det (B) + \operatorname{tr}(A\operatorname{adj}(B))$?
2
votes
1answer
59 views

Nicest operators on a vector space

Axler writes in his book that "nicest operators on $V$ are those for which there is an orthonormal basis of $V$ w.r.t which the operator has a diagonal matrix". i.e. orthonormal basis of $V$ ...
3
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0answers
34 views

Question regarding example of toric variety and generators of cone

Consider the canonical example taking n=2, and taking the cone $\sigma$ generated by the vectors $e_{2}$ and $2e_{1} - e_{2}$. The dual cone $\sigma^{v}$ is defined as the set of vectors in the dual ...
12
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9answers
2k views

Matrix with zeros on diagonal and ones in other places is invertible

Hi guys I am working with this and I am trying to prove to myself that n by n matrices of the type zero on the diagonal and 1 everywhere else are invertible. I ran some cases and looked at the ...
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2answers
49 views

Remembering the definition of the Jacobian: any tips?

I find it impossible to remember that the Jacobian of $f: \mathbb R^n \to \mathbb R^m$ is $$ \begin{pmatrix} {\partial f_1 \over \partial x_1} & {\partial f_1 \over \partial x_2} & \dots ...
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vote
1answer
45 views

Generate a random neutrally stable matrix

I need to generate random real matrices such that all eigenvalues have real part equal to 0 -- i.e. random neutrally stable matrices. What's the simplest way to do this? Note that I don't care about ...
4
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3answers
461 views

On Learning Tensor Calculus

I am highly intrigued in knowing what tensors are, but I don't really know where to start with respect to initiative and looking for an appropriate textbook. I have taken differential equations, ...
3
votes
2answers
89 views

Prove that the set of bases is linearly independent

Suppose that $W$ and $W'$ are subspaces of the vector space $V$ with the property that $W\cap W'=\{0\}$, and suppose that $\beta$ is a basis for $W$ and $\beta'$ is a basis for $W'$. Prove that the ...
0
votes
2answers
249 views

Give an example of a matrix A such that im(A) is the plane with the normal vector [1,3,2] in $R^3$

Give an example of a matrix A such that im(A) is the plane with the normal vector \begin{bmatrix}1\\3\\2\end{bmatrix} in $R^3$ How would I go about doing this question. The solution manual doesn't ...
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2answers
16 views

trying to solve a systems of equations with one inequality

I am trying to create a website that would run off this mathematical formula. I have tried to solve it but I got that there was no answer. I am only in pre-algebra and want a second opinion on if I ...
0
votes
2answers
43 views

Simultaneous equations with three parts

\begin{align*} 6a +24b +18c &= 168\\ 8a +28b +22c &= 208\\ 4a + 20b +20c &= 140 \end{align*} I've tried doing this multiplying so they cancelled out but I've always gotten decimal point ...
0
votes
2answers
228 views

derivative of matrix function with kronecker product

In the derivation of an estimator, I'm meant to find the minimum of the following matrix scalar function: $\underset\beta {argmin}$ $[S Y^\prime M^\prime - SX^\prime (kron(I_N,\beta) ) M^\prime ...
0
votes
2answers
32 views

Order of composition when dealing with transformations

I have been struggling with a question in my book. $T$ is a translation of $(+5,+4)$, $M$ is a reflection in the line $y=x$. $R$ is a 90 degree anticlockwise rotation about $(0,0)$ Write down ...
2
votes
0answers
57 views

cohomology of general linear group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $\mathrm{GL}_n(\mathbb{Z}_2)$ be the group consisting of all $n\times n$ matrices with entries in $\mathbb{Z}_2$ with non-zero determinant. What is the ...