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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3
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3answers
597 views

What is the dimension of this Grassmannian?

Why is $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension? Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all ...
5
votes
3answers
147 views

Show that $(1,x,x^2),(1,y,y^2),(1,z,z^2)$ form a basis of $\mathbb{R}^3$ iff $x\neq y, x \neq z, y \neq z$

I'm having some trouble with this one because I always get negated statements. If I try to prove both direction directly I get that three elements are all not equal to each other and the three vectors ...
0
votes
2answers
146 views

How can i figure out the points of a rectangle by just knowing the origin, width and length?

I've come across a mathematical problem, which I can't seem to solve with my limited geometry and trigonometry knowledge or by help of Wikipedia. I need to know the coordinate points of each corner ...
19
votes
3answers
1k views

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
14
votes
1answer
4k views

Effect of elementary row operations on determinant?

1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant results ...
1
vote
1answer
22 views

Finding the coefficients for a sparse matrix

I have a m*n matrix which i know is shaped like this : if a1, a2, ..., am, and b1, b2, ..., bn are real numbers, then for any matrix coefficient m(i,j) = ai*bj Preliminary question : how do you call ...
1
vote
1answer
629 views

Find orthogonal matrix $P$ such that $P^tAP = D$ for $A$ is a normal operator

For $$A = \;\;\; \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \\ \end{pmatrix} $$ there is a real orthogonal matrix $P$ such ...
2
votes
1answer
88 views

Sylvester matrix and GCD degree

How to prove that the degree of a $\gcd$ of two polynomials is equal to the dimension of the null space of the Sylvester matrix? I know that any linear combination of the rows of $S(u,v)$ is a linear ...
1
vote
2answers
133 views

Orthogonal complement of orthogonal complement

Let $S$ be some linear space. $S^{\perp}=\{y\;|\;\forall x\in S: \langle x,y \rangle=0\}$. So, how do I prove that $(S^{\perp})^{\perp}=S$ ?
1
vote
2answers
561 views

How do I find the Jordan normal form of a matrix with complex eigenvalues?

I'm trying to obtain the Jordan normal form and the transformation matrix for the following matrix: $A = \begin{pmatrix} 1 & 0 & 0 & 0 \\\ 1 & 0 & 0 & 1 \\\ 0 & 1 & 0 ...
0
votes
0answers
238 views

Strassen's Matrix Multiplication Example Problem

How to multiply two matrices using strassen's matrix multiplication.I have only learned the theory part but i cannot find any examples on the net. Could some one explain with two 2X2 Matrices.
2
votes
1answer
148 views

Every subspace of $\mathbb{R}^{n}$ has an orthonormal basis

Is there a non-constructive proof of this statement, i.e., one that avoids Gram-Schmidt?
-1
votes
1answer
267 views

Questins on Formulae for Eigenvalues & Eigenvectors of any 2 by 2 Matrix

Let $A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$. Then $\det(A - \lambda I) = 0 \implies \lambda_{\pm} \frac{\text{Trace} \pm \sqrt{\text{Trace}^2 - 4\det}}{2} = ...
4
votes
3answers
222 views

Diagonalizable matrices A and B with $\mathrm{Tr}(A^k)=\mathrm{Tr}(B^k)$ have the same characteristic polynomial? [duplicate]

Let $A$ and $B$ be $n \times n$ matrices with entries in a field F. Suppose $A$ and $B$ are diagonalizable in some extension field E of F and that $\mathrm{Tr}(A^k)=\mathrm{Tr}(B^k)$ for all integers ...
1
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0answers
23 views

Show C is not 1-error correcting by using Slepian decoding

Let C $\subseteq$ $ \mathbb{Z}_2^5$be a linear code with generator matrix $$G=\begin{bmatrix}1 & 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 1 & 1\\0 & 0 & 1 & 0 & ...
0
votes
1answer
143 views

Solution to a quadratic form

I'm trying to find a closed form solution of the following quadratic form for $x$. $x^{T}Dx = c$ where $c$ is just a constant placeholder for some terms on the other side. I know that, because $D$ ...
1
vote
2answers
22 views

Solving a system of linear equations?

Let $p$ be a particular solution to: $$Ax = b$$ Let $g$ be the general solution to the above equation. Let $h$ be the general solution to the equation: $$Ax = 0$$ Then $$g = p + h$$ Could ...
1
vote
2answers
89 views

Find bases for N(T) and R(T) and verify the Dimension Theorem

$T\pmatrix{a11&a12&a13\\a21&a22&a23\\a31&a32&a33}$= $\pmatrix{a11+a12+a13&a21+a22+a23\\a21+a22+a23&0}$ I know what the Dimension Theorem is and by looking at the ...
0
votes
2answers
62 views

Unknown linear transformation

Okay so I'm just prepping for my linear algebra final and I've come across questions like this before and I'm not sure how to solve. It seems pretty simple and I have an idea of what's going on but I ...
2
votes
5answers
3k views

Finding a 2x2 Matrix raised to the power of 1000

Let $A= \pmatrix{1&4\\ 3&2}$. Find $A^{1000}$. Does this problem have to do with eigenvalues or is there another formula that is specific to 2x2 matrices?
2
votes
1answer
116 views

If a linear transformation is similar to another, then they have the same eigenvalues.

I need help getting through this proof Let $T : R^n \rightarrow R^n$ be a linear transformation. T has an eigenvalue $\lambda$ if there exists some non-zero vector $\vec{x} \in R^n$ such that ...
2
votes
1answer
32 views

Explicit formula for a right splitting once we have a left splitting

Assume we have a short exact sequence (of abelian groups or vector-spaces, it doesn't matter) $$0\rightarrow A\stackrel{\iota}\rightarrow B\stackrel{\pi}\rightarrow C\rightarrow 0.$$ If we have a ...
1
vote
1answer
67 views

proving $W_1 + W_2$ is a subspace of V

I'm just trying to clarify one of my answers here. Here is the question: if $W_1$ and $W_2$ are subspaces of a vector Space $V$, show that $W_1 + W_2 = \{x+y : x \in W_1, y \in W_2\}$ is a ...
0
votes
0answers
96 views

How checking $(-1)^n$ for $H=H_{n+m}$ is equivalent to checking $(-1)^{m+1}$ for $H_{2m+1}$?

This is from "Mathematics from Economists" by Simon and Blume: To determine the definiteness of a quadratic form of $n$ variables, $Q(\mathbf{x})=\mathbf{x}^TA\mathbf{x},$ when restricted to a ...
1
vote
1answer
43 views

Inner product doubt

I have a doubt about a problem involving inner product spaces. The exercise is: Given the subspace generated by the vectors $ (1,1,1) $ and $ (1,-1,0) $, find the orthogonal subspace and give a ...
4
votes
1answer
246 views

Linear map $f:V\rightarrow V$ injective $\Longleftrightarrow$ surjective

Maybe I am not good at looking for the right questions but I haven't seen this task anywhere so I hope it is no duplicate. I have to prove the following statement: Let $V$ be a finite dimensional ...
2
votes
0answers
67 views

Translation invariance and finite dimension imply smoothness

Let $X$ be linear subspace of $C(\mathbb R)$, the set of continuous functions on $\mathbb R$, which is closed under translations, i.e., if $f\in X$ and $h\in\mathbb R$, then $\tau_h f\in X$, where ...
1
vote
2answers
359 views

Injective and surjective functions on a matrix

Suppose we have a function $G:M_2(\mathbb R) \to S_2(\mathbb R)$ where $S_2(\mathbb R)$ is a symmetric matrix such that $ S_2(\mathbb R) = \left\{A = \begin{bmatrix} a & b\\ c & ...
1
vote
2answers
53 views

Inner product space problem

First of all, I apologize for my English. I'm Spanish, so I hope you can all understand me. Here is my problem. Given the inner product: $$ \int_0^\pi f(x)g(x)dx\ $$ in the space of continuos ...
1
vote
2answers
123 views

Symmetrical endomorphisms and quadratic forms

(This last part of my linear algebra course is causing me quite a bit of headaches, so please be patient) Let $V$ be a vector space over the real field, and we'll indicate with $(\cdot,\cdot)$ its ...
0
votes
0answers
17 views

Sum of transformations - consequences

Well, in matrix-vector transformation, I know that if we take the composition of a transformation, like: $T(x) = A\cdot x$ and $T$ maps from $R^n$ to $R^m$ $S(x) = B\cdot x$ and $S$ maps from $R^m$ ...
1
vote
1answer
83 views

Prove this isomorphism of $K$-algebras

Seen this is a lot of literature, usually without proof. Was just wondering how is it: $A$ is a $K$-algebra, where $K$ is a field. Then $A \otimes_K A^{op} \simeq M_r(K)$ where $r=\dim(A)$. I ...
3
votes
3answers
60 views

Gaussian elimination - number of solutions

How do I know how many solutions does a system of linear equations have? I have such system of linear equations: $\begin{cases} (1+b)x+y+z=1\\ x+(1+b)y+z=b\\ x+y+(1+b)z=b^2 \end{cases}$ And I would ...
3
votes
3answers
174 views

Diagonalization of linear operators

First of all, I´m sorry for my English, I´m Spanish so I hope you can all understand me. Here is my problem. Let $T(p(x))=p(x+1)$ be a linear operator from the space of ...
1
vote
1answer
141 views

If $A$ has two eigenvalues $\lambda _1, \lambda_2$ and $\dim (E_{\lambda_1})=n-1$, then $A$ is diagonalizable

Suppose that $A \in M_{n\times n}(\Bbb F)$ has two distinct eigenvalues $\lambda_{1}$ and $\lambda_{2}$ and that $\dim (E_{\lambda_1})=n-1$ show that $A$ is diagnolizable.
6
votes
1answer
98 views

Maximal value of dimension [closed]

I'm stuck on a question if you can help me : Show that the maximum dimension of a subspace of $\mathcal M_n (\mathbb F)$ not containing an invertible matrix is $n (n-1)$.
0
votes
1answer
88 views

Eigenvalue decomposition of $D \, A \, D$ with $A$ symmetric and $D$ diagonal

Let $A$ be a real, symmetric matrix. It admits the eigenvalue decomposition $A = U \Lambda U^T$ where the eigenvectors are chosen to be orthogonal. Let $D$ be a diagonal matrix and $B = D A D = D U ...
4
votes
1answer
103 views

Is there a more conceptual proof of this fact?

Equip ${\mathbb R}^3$ with the usual scalar product $(.|.)$. Let $A$ be the matrix $$ A= \left(\begin{matrix} 1 & 2 & 3 \\ -2 & 4 & 5 \\ -3 & -5 & 6 \\ \end{matrix}\right) $$ ...
0
votes
1answer
36 views

Determinant: Effect on changing entry in a matrix on Cofactor Expansion [GStrang 5.2.22]

I can't discern why $N_n = S_n - S_{n - 1}$ from the answer (I define these below). I computed: $S_1 = 3, S_2 = 8$. By cofactor expansion along the $3$rd column (WLOG, I could've picked row): $S_3 = ...
3
votes
1answer
37 views

matrices problems, unclear concept need explanation.

$$I.A \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ \end{pmatrix}=\begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix} $$ $$II.A ...
1
vote
0answers
21 views

proving there exist another basis of non-degenerate quadratic space (V,B) other than the given basis

If {$v_i$} is a basis of non-degenerate quadratic space ($V,B$) (finite), prove that there exists another basis {$w_i$} such that $$B(v_i,w_j)=1 (i=j)$$ $$or 0(i \neq j)$$ Sorry for the ugly text ...
1
vote
0answers
32 views

Nonlinear animation algorithm

I'm a programmer working on an animation algorithm. Heuristically, it starts out like this: ...
0
votes
3answers
169 views

Find the determinant, assuming that

Given that $$\begin{vmatrix} a & b & c \\ d & f & g\\ q & w & e \end{vmatrix} =5$$ It is a whole matrix above. $$\begin{vmatrix} a & b & c \\ d & f & ...
0
votes
1answer
425 views

Find the Equation of the Plane Containing the Point $ (2, 3, -2)$ and the Line$\frac{ x-1}{6} =\frac{ y+1}{2} = z-3$

A certain plane contains the point $P=(2, 3, -2)$ and the line $\frac{ x-1}{6} =\frac{ y+1}{2} = z-3$. I know that to find the equation of a plane I must have a point and a vector normal to that ...
0
votes
1answer
72 views

Proving that self-adjoint operators have only real eigenvalues

Does anyone know how to prove that if $T$ is a self-adjoint operator, then all of the eigenvalues of $T$ are real?
2
votes
2answers
132 views

Find eigenvectors of an infinite dimensional vectorspace Pn(R)?

Define a linear operator T:Pn(R)--->Pn(R) by T(p(x))=p(x)+p(2)x. (a) How many distinct eigenvalues does T have? (b) What are the dimensions of their corresponding eigenspaces? At first I started ...
1
vote
2answers
107 views

Fredholm integral equation of first kind

I want to solve the Fredholm integral equation of first kind: $$ \int_L K(x,y)U(y)dy = f(x) $$ in these equation the function $U(y)$ is the unknown and the so-called kernel $K$ and the right hand side ...
1
vote
4answers
88 views

what are the eigenvalues in orthgonal matrix, How to explain?

what are the possible eigenvalues of an orthogonal matrix? I got the answer key which says its 1 and -1 but it doesn't explain well
0
votes
1answer
35 views

Determinant Algebra computing

Let $A, B \in M_3 (\mathbb{R})$, two invertible matrices such that $B^TA^{-1}= 2I_3$ and $ABA^T= I_3$. How can I prove that $\det A + \det B = 9/2$? Thanks!
2
votes
1answer
89 views

Show there exists an invariant subspace $W\subseteq \mathbb R^n$..

Let $T$ be an orthogonal operator in $\mathbb R^n$. How can I show there exists in invariant subspace $W\subset \mathbb R^n$ such that $\textrm{dim}(W)=1$ or $\textrm{dim}(W)=2$? There was a hint for ...