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 ...

learn more… | top users | synonyms

0
votes
1answer
23 views

What can we say about output of Gram–Schmidt process

Given $\{x_1, \dots, x_{n-1}\}$ linearly independent vectors and $x_n \in \operatorname{span}\{x_1, \dots, x_{n-1}\}$ and let $\{\hat{x_1}, \dots, \hat{x_{n-1}}, \hat{{x_n}}\}$ be the output of the ...
3
votes
1answer
36 views

Does every isomorphism between $V$ and $V^*$ send some basis to its dual basis?

Suppose that I have a vector space isomorphism $\theta: V \to V^*$ where $V$ is any vector space (probably over $\mathbb{C}$ is required) and $V^*$ is its dual space. Is it always possible to find a ...
2
votes
1answer
21 views

Eigenvalues of hermitian plus skew-hermitian PSD matrix

I was wondering, suppose you have a matrix of the form $A=B+iCC^\dagger$ where $^\dagger$ denotes the hermitian conjugate. $B$ is hermitian and $CC^\dagger$ is obviously hermitian positive ...
0
votes
2answers
20 views

Orthographic projection of point [0, 0, 0]

What is the easiest way to calculate orthographic projection of point $[0, 0, 0]$ on a plane given by formula $x - y + z = 1$?
9
votes
3answers
84 views

Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series

Let $T: V \to V$ be any linear transformation on a real or complex vector space $V$. Show that there exists $\epsilon_0 > 0$ $($depending on $T$$)$ so that $I + \epsilon T$ is invertible for any ...
1
vote
1answer
39 views

How can a $k\times (k-m)$ matrix be multiplied by a $k\times m$ matrix?

While reading a book on differential geometry, I came across this line: Since the differential $d\psi_0(x_0):\mathbb R^m\to \mathbb R^k$ is injective, there is a matrix $B\in \mathbb R^{k\times ...
0
votes
0answers
19 views

Finding basis of inverse image

Let $\psi $ be a linear transformation such that$$\psi ([x_1,x_2,x_3,x_4])=[x_1+x_3+x_4, -x_2-x_4,x_1+x_2+x_3+2x_4].$$ Find basis of inverse image $\psi^{-1}(W)$ of subspace ...
-1
votes
1answer
23 views

Is $\frac{F(b)-f(a)}{b-a}x < f(x)$ for $x\in[a,b]$? [on hold]

As stated in the description, I want to know whether the following statement is true or not Is $\frac{F(b)-f(a)}{b-a}x < f(x)$ for $x\in[a,b]$?
0
votes
2answers
42 views

Why is this map not surjective at the origin?

$f:\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ $f(x) = |x|^2$ Then the derivative map is $df_{x}(v)=2\sum_{i}{x^iv^i}$ is surjective except at 0. Is it because at 0 df only goes to 0, and doesn't ...
-2
votes
1answer
29 views

Find basis of an image

Let $ \psi : \mathbb{R}^4 \rightarrow \mathbb{R}^3$ be a linear transformation described by a formula $$\psi ([x_1,x_2,x_3,x_4])=[x_1+x_3+x_4, -x_2-x_4,x_1+x_2+x_3+2x_4].$$ Find basis of image ...
-1
votes
0answers
21 views

Isomorphism type of a finite matrix group

Give a known group to which $SO_5$ is isomorphic. Where $SO_5$ consists of $$\begin{pmatrix} \cos\frac{2k\pi}{5} & \sin\frac{2k\pi}{5} \\ -\sin\frac{2k\pi}{5} & \cos\frac{2k\pi}{5} ...
0
votes
0answers
27 views

Physical or geometric meaning of the trace of a matrix

The geometric meaning of the determinant of a matrix as an area or a volume is dealt with in many textbooks. However, I don't know if the trace of a matrix has a geometric meaning too. Is there ...
0
votes
0answers
12 views

Direct sum of two spaces

Let $\alpha_1=[1,1,0,1]$, $\alpha_2=[1,0,1,1], \alpha_3=[1,1,1,1],\alpha_4=[0,1,1,1]$ be a vectors from $\mathbb{R}^4$ let $U=span(\alpha_1, \alpha_2) \ and \ W=span(\alpha_3, \alpha_4)$ Check that ...
1
vote
1answer
54 views

Determinant: Continuity?

Building on the previous thread: Determinant: Definitions? Presumptions: Differential Geometry, Functional Analysis Given a vector space $V$. Consider an endomorphism $T:V\to V$. Define its ...
0
votes
0answers
20 views

Where can Gaussian Elimination be used?

I have searched for this and came to know about it that it is traditionally used to solve linear equations, finding determinant, rank of matrix, inverse of matrix. There was a problem on codechef: ...
0
votes
1answer
15 views

a question about general and particular solutions

We have a $3\times 6$ matrix $A$ with rank $3$ (this is all the information we have, no matrix given). Here comes the questions: What is the number of free variables in the solution to the system ...
2
votes
1answer
29 views

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,.,1,0,.)$ and a bounded linear functional $\Phi$ find $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges?

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,...,1,0,...)$ and a bounded linear functional $\Phi$ find a value of $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges for $b_n=\Phi(e_n)$? Ok so since ...
1
vote
1answer
10 views

Odd coefficient in $M\in \mathcal{M}_n(\Bbb{Z})$ satisfies $n\le m\le n²-n+1$.

Let $M\in \mathcal{M}_n(\Bbb{Z})$ I would like to prove that all odd coefficient of $M$ satisfies $n\le m\le n²-n+1$. In fact I don't see why $m$ is necessary bigger than $n$. I can only prove ...
0
votes
0answers
11 views

Prove existence of (Nash) equilibrium

My question is about proving the existence of Nash equilibrium for a game involving two players. $x$ is player 1's strategy and $y$ is player 2's strategy; both strategies are continuous. For each ...
2
votes
2answers
64 views

What is a nice definition of the determinant not relying on representations by matrices?

Foundation for: Determinant: Continuity Presumptions: Differential Geometry, Functional Analysis Given a vector space $V$. Consider an endomorphism $T:V\to V$. The rank of an endomorphism: ...
0
votes
2answers
18 views

Epimorphism of linear transformation

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=[x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4] $ When this transformation is epimorphic i.e. what ...
0
votes
2answers
22 views

Show that $trv=\lim_{t\to 0}\frac{\det(I+tv)-1}{t}$ for any n by n matrix

Prove that for any n by n real matrix $v\in {\mathbb R}^{n\times n}$, $trv=\lim_{t\to 0}\frac{\det(I+tv)-1}{t}$, where $t\in\mathbb R$, $I$ is the identity matirx, and $trv$ denotes the trace of ...
1
vote
1answer
18 views

Eigenvalue of altered matrix: $pI_n + qA$

As a part of an exercise I have to prove the following: Let $p,q \in \mathbb{R}$. Let $A$ be an $(n \times n)$ matrix. Let $I_n$ be the $(n \times n)$ identity matrix. If $A$ has an eigenvalue ...
0
votes
1answer
16 views

Determining transformation matrix from six points

Given that I have the locations of three points: p1 = [1.0,1.0,1.0] p2 = [1.0,2.0,1.0] p3 = [1.0,1.0,2.0] ...and I know their transformed counterparts: ...
0
votes
0answers
12 views

Lagrange Newton Method Singular Matrix

i implemented the lagrange-newton method in python to find the problem to nonlinear optimizing problem for learning purposes. But every guess i made a guess for the initial values the resulting ...
1
vote
0answers
22 views

A basis that simultaniously diagonalizes two matrices?

Given matrices $A$ and $B$, assuming they can be diagonalized with the same S, so that $D_A = SAS^{-1}$ and $D_B = SBS^{-1}$ ... how would one find the basis that makes up $S^{-1}$? I've got two ...
3
votes
6answers
534 views

How to solve these equations for x and y..

equations are $(x-y)(x+2y)(2x+y) = 20$ and $x^2+xy+y^2 = 7$ i want the METHOD not the solutions
0
votes
1answer
13 views

How to intersect row and column sub-spaces?

What is the connection or, intersection between row space and column space of a square matrix? how can I intersect two different sub-spaces?
0
votes
0answers
11 views

Interpolating transformation matrices

I read not to interpolate transformation matrices by linearly interpolating. Can someone explain to me why interpolating transformation matrices by linearly interpolating the matrix components is a ...
2
votes
1answer
29 views

Faulty proof that $V=U_1 \oplus W$ and $V=U_2 \oplus W$ implies $U_1 = U_2$

The question is as follows: Prove or give a counterexample: if $\ U_1, U_2, W$ are subspaces of $V$ such that $V=U_1 \oplus W$ and $\ V = U_2 \oplus W$, then $\ U_1 = U_2$. I happily ...
0
votes
1answer
47 views

Describe the following set

We are supposed to describe the set $\bigcup_{n=1}^\infty A_n$ with a proof. $A_n = \{(x, y) \in \mathbb{R}^2 | y-x^{2n} \geq 0 \}$ I get that the set contains all functions where $y\geq x^{2n}$ ...
0
votes
1answer
38 views

Find a minimal spanning set of a set of matrices

I'm supposed to find a minimal spanning set of $W = \{A \in M_n(\mathbb{R}) | \operatorname{Tr}(A) = 0\}$ First of all, what is a minimal spanning set? I can't find the term anywhere in the notes my ...
0
votes
2answers
51 views

Prove there exist an isomorphism

Let $V$ be a vector space and $U,W,Z$ are subspaces of $V$, where $V=Z \oplus W=Z \oplus U$ Prove there exist linear isomorphism $f:V \to V$ such that for every $\gamma \in Z, \ \ f(\gamma)=\gamma$ ...
0
votes
2answers
23 views

Linear composition

can you help me with this quest? About composition $f$ and vector space $\mathbf{V}=\mathbb{Z^4_2}$ we know the following: $f \circ f = id_V$,$~~f $ $ \left(\begin{array}{ccc} 1\\ 0\\ 1\\ 0\\ ...
2
votes
2answers
59 views

Show that a basis change is a linear transformation [on hold]

Show that the map v = [v]E → [v]B for all v ∈ Rn defines a linear transformation TB : Rn → Rn. B = {b1, b2, b3,...,bn} is a basis of Rn. Any vector v ∈ Rn can be uniquely expressed as a linear ...
0
votes
0answers
36 views

Orthogonal vectors to Bivector

If we have set of orthogonal vectors (X) and form that set we create a set of bivectors (Y). Is there a relation among Y based on X? Can we say that Y's are orthogonal as well?
0
votes
1answer
29 views

Prove that the set of points that make up the unit circle are uncountable

My math teacher asked us to prove that the set of points that make up the unit circle are uncountable. We are supposed to do this by "exhibiting" (not sure if this means it can be a proven through ...
1
vote
0answers
35 views

equivalent inner-product vector for one

I have a map that projects a $k$ dimensional vector $x$ to an $m$ dimensional vector $\phi(x)$. The vector function (map) $\phi$ can be any linear or non-linear function of $x$, which is not ...
0
votes
0answers
11 views

Show that there exists a Hermitian form of signature $(p,q)$.

Let $K = \mathbb{Q}(\sqrt{-2})$ with $V_K = K^n$ considered as a $K$-vector space. Suppose $p, q \in \mathbb{Z}_{>0}$ such that $p + q = n$. Show that for any such $p$ and $q$ there is a Hermitian ...
1
vote
2answers
23 views

Prove vectors create a basis

Let $V$ be a vector space and $U,W,Z$ be it's subspaces where $V=Z \oplus U=Z\oplus W$. We know that $\beta_1,...,\beta_k$ is a basis of $U$ and $\beta_i=\gamma_i+\delta_i$ where $\gamma_i \in Z$ and ...
0
votes
2answers
30 views

Let $V=\{f \in X \mid f(0)=f(1)=0\}$ be a linear subspace of $C[0,1]$. Show $(V,\|\cdot\|_\infty)$ is Banach.

Can you please confirm if my proof is correct and if not show where I went wrong. Thanks! Let ${f_n}$ be a Cauchy sequence in $V$ then $f_n(x)$ is a real number for each $x\in [0,1]$ Hence ${f_n(x)}$ ...
26
votes
14answers
2k views

What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside ...
0
votes
4answers
42 views

Proving something is a linear transformation? [on hold]

If $T:\Bbb R^3 \to \Bbb R^2$ given by $$ T\begin{bmatrix} a\\ c\\ e\\ \end{bmatrix}= \begin{bmatrix} c\\ a\\ \end{bmatrix} $$ then $T$ is a linear transformation: true or false, ...
0
votes
1answer
21 views

The convergence of $f_n(t)$ to $f_n$ in the supremum norm implies $f_n(t)\rightarrow f(t)$ as $n\rightarrow\infty$?

I'm awful at these problems so I was just posting this to confirm whether my solution is correct. As: $$(f_n)_{n\geq 1} \in X \rightarrow f \in X \mbox{ in the supremum norm}$$ $$\mbox{For all } ...
4
votes
1answer
46 views

General linear group and special linear group

Consider the general linear group $$GL(n,\mathbb R)=\{g\in {\mathbb R}^{n\times n}\mid\det(g)\neq 0\}$$ Prove that the derivative of the function $f=\det:{\mathbb R}^{n\times n}\to\mathbb R$ is ...
1
vote
1answer
21 views

Examples of Unitary Matrices with coefficients all having the same amplitude

I am looking for examples of unitary matrices like this one $$A = \frac{1}{\sqrt{2}}\left( \begin{array}{rr} 1 & 1 \\ 1 &-1 \end{array} \right)$$ where each coefficient has the same amplitude, ...
1
vote
0answers
16 views

Phase Portrait of DE's

How would I graph the phase portrait of $$ x' = x^2+y^2-2 \qquad y' = y-x^2 $$ ? Could someone provide some insight by hand or perhaps a computer-generated image?
1
vote
0answers
21 views

Give the following linear transformation find values of parameter

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4 $ When this transformation is epimorphic i.e. what ...
1
vote
2answers
40 views

let A be a $2\times 2$ matrix . Then the smallest number $n\in \mathbb N$ such that $A^n=I$ is

let A be a $2\times 2$ matrix $\begin{pmatrix} \sin \frac \pi {18} & -\sin \frac {4\pi} {9}\\ \sin \frac {4\pi} {9}&\sin \frac \pi {18}\end{pmatrix}$. Then the smallest number $n\in \mathbb N$ ...
0
votes
3answers
35 views

Abstract Linear Transformation Question

I had this question on a quiz today and no idea how to solve it. Please help. Let $ T: \mathbb{R}^n\rightarrow \mathbb{R}^n $ a linear transformation defined by: $\forall \begin{bmatrix}x_1\\x_2\\ ...