# Tagged Questions

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

4answers
224 views

### Linear Algebra and Applications [duplicate]

Possible Duplicate: Where to start learning Linear Algebra? Does anybody know of a good book in Linear Algebra for self study for a beginner? I look for a book with many exercises.
0answers
16 views

### Does convex and radially open imply open?

I want to show that a convex set $A$ is radially open iff $A\cap W$ is open in W, for every finite dimensional linear subspace. Here the 'openness' we are talking about is from any normed space. ...
1answer
37 views

### Groups of transformations

I tried to find literature and articles about groups of transformations, but mostly what I found is either about groups or about transformations. Can you suggest me literature where groups of ...
2answers
47 views

### self-adjoint transformation.

I have matrix that represent $T:V \to V$ (linear map over $\mathbb{R}$) according to basis $E$. $E$ is not an orthonormal set. how can I know if this $T$ is self-adjoint. I know that if $E$ was ...
1answer
18 views

### How to show that a valid inner product on V is defined with the formula $[x, y] = \langle Ax, Ay\rangle$?

Let $A \in L(V,W)$ be an injection and $W$ an inner product space with the inner product $\langle \cdot,\cdot\rangle$. Prove that a valid inner product on $V$ is defined with the formula \$[x, y] = ...
1answer
32 views

### Arrangements of affine hyperplanes

Fix $n>0$ and $X\subseteq\mathbb{R}^n$. Call a function $f:X\longrightarrow \mathbb{R}$ linear if it is of the form $$f(\bar{x})=a_1x_1+\ldots+a_nx_n+b$$ for some $a_i,b\in\mathbb{R}$. Now ...
0answers
41 views

### Given a vector space $V$, show that the following statements are equivalent.

Given a subset $W$ of $V$ then I want show that, $\forall v \in V, w \in W$ $\exists \lambda \in \mathbb{R}$ such that $w + \alpha v \in W$ for any $0 < \alpha < \lambda$ iff \$\forall v \in ...
3answers
1k views

### Does non-symmetric positive definite matrix have positive eigenvalues?

The title says it. I found out that there exist positive definite matrices which are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues. Does this hold for ...
1answer
108 views
+100

### Properties of the Cone of Positive Semidefinite Matrices

The set of positive semidefinite symmetric real matrices form a cone. We can define an order over the set of matrices by saying $X\geq Y$ if and only if $X-Y$ is positive semidefinite. I suspect that ...
1answer
40 views

### Is the inverse function smooth?

Imagine that we have a function $Inv$ that maps $A \rightarrow A^{-1}$, where A is an invertible square matrix. now my questions is: how do i see that this function is arbitrarily often ...
2answers
44 views

### Cases where characteristic and minimum polynomial coincide

Given a matrix such as $$\pmatrix{0 & 0 & 2 \\ 1 & 0 & -1 \\0 & 1 & 1 \\ },$$ whose characteristic polynomial is $-X^3+X^2-X+2.$ How it could be deduced that it equals ...
1answer
21 views

### how would $f(T)$ look like if …

I know the result that if $T:V_F\to V_F$ is a linear operator then for any polynomial $f(x)\in F[x],~f(T)$ is a linear operator. Now my question is how would $f(T)$ look like if $f(x)$ is the zero ...
1answer
39 views

### Fast way to calculate Eigen of 2x2 matrix using a formula

I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
1answer
30 views

### calculate kernels of matrices with angles

So my professor gave me this question: I have to find the basis of the eigenvalues of this matrix \begin{pmatrix} \cos(q) & \sin(q)\\ \sin(q) & -\cos(q)\\ \end{pmatrix} so I calculate ...
2answers
106 views

### How to show that there exists a scalar $\lambda$ so that $A = \lambda I$? [duplicate]

Let $V$ be a finite dimensional vector space, $V \ne \{ 0 \}$ and $A\in L(V)$ an operator which commutes with every operator from $L(V)$. Show that there is a scalar $\lambda$ so that the following ...
1answer
21 views

### Some remarks/questions from Primary Decomposition Theorem to get verified

In course of self-studying the Canonical Form in Linear Algebra I'm trying to put some remarks from the concept I acquired from Primary Decomposition Theorem which reads as follows: Let \$T:V_F\to ...
1answer
26 views

### Consequences of a rectangular matrix being of maximal rank

I have a real matrix $A$, $(m+1) \times m$ and a vector $b \in \mathbb R^{m+1}$ such that $b_{m+1}=0$. For any vector $u\in \mathbb R^m$, $Au=0 \Rightarrow u=0$. This means that $A$ is a rectangular ...
3answers
48 views

### Vectors that form a triangle!

I have a problem here. How can I prove that sum of vectors that form a triangle is equal to 0 (AB+BC+CA=0) ? Thank you!
1answer
16 views

### (Follow Up) Checking the solutions of a quadratic polynomial

I'm following up from this question: Solve a polynomial involving geometric progression? I have had trouble with this question: "Solve the equation $8x^3−38x^2+57x−27=0$" if the roots are in ...
1answer
32 views

### Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$

Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$ Using $\det(Q)\det(\bar{Q}^T) = I$, I get to the stage $\det(\bar{Q})\det(Q)=1$, but can't do much else with it. Thanks for ...
1answer
708 views

### Proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ in Penrose graphical notation

For two matrices $\textbf{S}$ and $\textbf{T}$, a proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ is given below in the diagrammatic tensor notation. Here $\det$ denotes the ...
1answer
32 views

### Characteristic polynomial of the differentiation map

Determine the characteristic and minimum polynomial of the differentiation map $D: \mathbb{R_n}[X]\longrightarrow \mathbb{R_n}[X]$ (where $\mathbb{R_n}[X]$ is a set of polynomials of degree at most ...
2answers
33 views

### How to prove two polynomials have no zeroes in common?

The question asked: Divide the polynomial $P(x) = x^3 + 5x^2 - 22x - 6$ by $G(x) = x^2 - 3x + 2$. I did, and got the answer: $(x+8)(x^2-3x+2)-22$. However, it now asks to: "Show that $P(x)$ and ...
1answer
212 views

### Property of partial traces

Consider the Kronecker product of $A \in M_m, B \in M_n$: $A \otimes B = \left( \begin{matrix} a_{11}B&...&a_{1m}B\\ \vdots&\ddots\\a_{m1}B&...&a_{mm}B \end{matrix} \right)$ \$A ...
3answers
21 views

### How to form a cubic equation with the substitution method?

I had this question: "Find the cubic equation whose roots are twice the roots of the equation $3x^3 - 2x^2 + 1 = 0$" In my first attempt, I solved it through the use of simultaneous equations, where ...
1answer
35 views

### Dimensions of vector subspaces in a direct sum are additive

$V = U_1\oplus U_2~\oplus~...~ \oplus~ U_n~(\dim V < ∞)$ $\implies \dim V = \dim U_1 + \dim U_2 + ... + \dim U_n.$ [Using the result if $B_i$ is a basis of $U_i$ then $\cup_{i=1}^n B_i$ is a basis ...
0answers
36 views

### On integral of a function over a simplex

Help w/the following general calculation and references would be appreciated. Let $ABC$ be a triangle in the plane. Then for any linear function of two variables $u$.  \int_{\triangle}|\nabla ...
1answer
94 views

### Possibly Wrong (Perhaps in the Right Direction Though)

I saw this question Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $B ∈ \mathbb{Q}^m$. Suppose that the system of linear equations $AX = B$ has a solution in $\mathbb{R}^n$. Does it necessarily have ...
4answers
61 views

### if $\mathbf x$ is sampled randomly from a hypercube on $R^n$, what is the probability density for $|\mathbf x| = d$

if the vector $\mathbf x$ is sampled randomly from a uniform distribution on $[0, 1]^d$, what is the probability density function for $|\mathbf x|$? Is it easy to scale for $[0, n]^d$?
2answers
32 views

### Linear Transformation Orthogonality

True or False: If $T$ is a linear transformation from $R^n$ to $R^n$ such that $$T\left(\vec{e_1}\right), T\left(\vec{e_2}\right), \ldots, T\left(\vec{e_n}\right)$$ are ...
1answer
49 views

### Is it possible that $V=\cup_{i=1}^n V_i?$

Let the vector space $V$ be decomposable into the non-zero subspaces $V_i;~i=1(1)n.$ Is it possible that $V=\cup_{i=1}^n V_i?$
1answer
41 views

### A question on linear operators

This is a problem I’ve been working on as part of my studies for an upcoming comprehensive exam: Let $F$ be a field, let $V\in F$-$\mathrm{Mod}$ be a finite-dimensional left $F$-vector space, and let ...
2answers
24 views

### Matrix involving values of polynomials

I've been doing this problem but im stuck. Be $f_1 f_2 f_3 \in \mathbb{R}_2$[$x$]. Proove that {$f_1$,$f_2$,$f_3$} form a base of $\mathbb{R}_2$[$x$] as $\mathbb{R}$ vector space, if and only ...
0answers
36 views

### Extending a rational entry matrix to an orthogonal matrix.

Suppose $M \leq N$ are positive integers and let $r_1,\ldots,r_M$ represent the rows of an $M\times N$ matrix $A$ in which: (i) the rows of $A$ are orthogonal and having the same norm, (ii) the ...
2answers
27 views

### Linear Algebra determinant and rank relation

True or False? If the determinant of a $4 \times 4$ matrix $A$ is $4$ then its rank must be $4$. Is it false or true? My guess is true, because the matrix $A$ is invertible. But there is ...
1answer
46 views

### What are the two main ways to prove that a matrix $\Bbb R^{n\times n}$ is definite positive.

What are the two main ways to prove that a real $n\times n$ matrix is definite positive? Is the first way: If a matrix is $n \times n$ symmetric matrix, then the associated quadratic form is ...
1answer
48 views

### Quadratic Equation with “0” coefficients

Let's say I have two objects $x$ and $y$ whose position at time $t$ is given by: $$x = a_xt^2+b_xt+c_x \\ y= a_yt^2+b_yt+c_y$$ And I want to find which (if any) values of $t$ cause $x$ to equal ...
1answer
37 views

### Sufficient condition for equality with order hypothesis

I have this very simple equation for arbitrary $a,b,c,d,e,f\in\mathbb{R}$ with $\mathbb{R}$ endowed with the usual order : $$a+b+c=d+e+f$$ So, if this equality is verified, I know that ...
3answers
199 views

### Riemann sphere and Maps

Could somebody please clarify the following for me? I am not too clear about the relationship between the Riemann sphere and Mobius maps. I know that we can through projection make some Mobius maps ...
1answer
58 views

### Matrices that satisfy $A^2,B^2,C^2$ and commute

Is there a set of matrices that satisfy all of the following constraints? 1) $A^2=0, B^2=0, C^2=0...$ where $A,B,C,D..$ are different matrices. 2) All of them commute. Edit: 3) \$AB \neq 0, AC \neq ...
1answer
18 views

### Finite difference method stability

I have shown that a finite difference method satisfies $$\underline{u}^{n+1}=((1+6\mu)\mathbb{I}-36\mu A^{-1})\underline{u}^n$$ I don't think that the rest of the question is necessary but it is ...
2answers
88 views

### How to show that if $p(A) = 0 \implies p(\lambda_0)=0$?

Let $V$ be a finite dimensional vector space and $V \ne \{ 0 \}, A\in L(V), \lambda_0 \in \sigma(A)$. If $p(\lambda)$ is an arbitrary polynomial for which the following applies: $p(A) = 0$, prove ...
0answers
29 views

### Eigenbasis of a Hilbert space: isomorphism

Let $K$ be a matrix containing the dot product between points in a Hilbert space $\mathcal{H}$ (assume that it is finite-dimensional). Then, we could form a basis using the eigenvectors of a normal ...
2answers
25 views

### Showing set of tensored states span a space

I have the four states  \lvert1\rangle \lvert1\rangle - \lvert0\rangle \lvert0\rangle \\ i\lvert1\rangle \lvert1\rangle + i\lvert0\rangle\lvert0\rangle \\ \lvert0\rangle \lvert1\rangle + ...
1answer
27 views

### The relationship between plane curves and the derivative of the Wronskian

I have found a theorem but I did not understand the proof. I'm looking for a clarification of the proof or a different proof. Let $f_1, f_2, f_3$ be the three components of a curve in $R^3$ ...
2answers
46 views

### Smooth maps on a manifold lie group

 \operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\ \begin{align} &n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\ &n = 2, \operatorname{GL}_n(\mathbb ...
4answers
304 views

### The path of complex structure.

$J = \left( {\begin{array}{*{20}{c}} 0&{ - {I_n}}\\{{I_n}}&0\end{array}} \right)$. If $I(t)$ is a path in $\rm M(2n, \mathbb R)$ ($2n \times 2n$ real matrix) such that $I(0)=0$ and ...
0answers
12 views

### Conditions for matrix operator to preserve complex symmetry on DFT vector?

Suppose there is a DFT vector $\mathbf{X}$ (complex vector) with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. ...
1answer
30 views

### Why don't we consider the zero subspace (which is readily $T$-invariant) in the definition of direct sum of linear operators?

Why don't we consider the zero subspace (which is readily $T$-invariant) in the definition of direct sum of linear operators? REF: Schaum's Outline of Linear Algebra
1answer
32 views

### Book on quadric surfaces with linear algebra

Most information that I can find about quadric surfaces is written from a calculus perspective - without using any matrices or vectors. However, I would like to have a reference that tells me the ...
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 151, "mathjax_display_tex": 7, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9101342558860779, "perplexity_flag": "head"}