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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Prove that $b_{\perp}^{T}b_{\parallel}=0$

If $A \in \mathbb{R}^{mxn}$ then the unique expansion of every $b \in \mathbb{R^{m}}$ is $b =b_{\perp}+b_{\parallel} $. Prove that $b_{\perp}^{T}b_{\parallel}$. Comment: Saying that they are ...
2
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1answer
161 views

In $\mathbb{R}^3$, if $v$ is orthogonal to $x$ and $y$, then $x \times y$ is a scalar multiple of $v$. [on hold]

Let $x, y, v \in \mathbb{R}^3$. If $v\neq0$ is orthogonal to $x$ and $y$, then $x \times y$ is a scalar multiple of $v$. We can do $$v\times(x\times y)=(v\cdot y)x-(v\cdot x)y=0$$ so, by ...
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0answers
55 views

How to reach Moore-Penrose pseudoinverse solution to minimize error function

Edit I'm trying to figure the derivation of the Moore-Penrose pseudoinverse for linear regression. The starting expression is the standard error function. I'm not quite sure how to expand on this ...
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1answer
41 views

What are the irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$? [on hold]

Find with proof all irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$.
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1answer
30 views

$V^{\oplus3}$, linear constraints. [on hold]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$, and let $W = V \oplus V \oplus V$. Prove that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
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1answer
32 views

How to determine the signs of the eigenvalues of a symmetric $3\times 3$ matrix?

This is a homework problem: Let $a,b,c$ be positive real numbers such that $b^2+c^2<a<1$. If $A=\begin{pmatrix} 1&b&c\\b&a&0\\c&0&1\end{pmatrix}$, then which of the ...
3
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1answer
42 views

Name of Inequality

Let $x_i, y_i$ be complex numbers for all $i$. Is there a name for the following inequality? $$\left| \sum_{i=1}^n x_i \right| \leq \sum_{j=1}^n |x_j| $$ In particular, is it a special case of this ...
2
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0answers
17 views

Matrix properties polylinear function [on hold]

the polylinear antisymmetric functions works on the rows of a matrix n by n and also if we define what it is we can easily connect it to the determinant but what is the step to connect the polylinear ...
2
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1answer
20 views

Finding same-vectors that have same coordinates in two different basis

I have two different vector basis: Default: $\{e_1,e_2,e_3\} = \{(1,0,0);(0,1,0);(0,0,1)\}$ Special basis: $\{e'_1,e'_2,e'_3\} = \{(1,1,1);(1,0,1);(0,2,1)\}$ My question is: How do I find which ...
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1answer
35 views

Polynomial ring: Direct sum of modules?

I got the following task from my professor and I wanted to ask for advice from you. Task: $K$ is a field I shall prove the following statement: $n \neq v$, $I_n + I_v = K[X]$ and is this a ...
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0answers
18 views

Transformation into a field with the result being a multiple of the determinant

Let $K$ be a field, $n \in N$ and d: $M_{n,n}(K) \to K $ an homogeneous and skew invariant transformation where $M_{n,n}(K)$ are the matrices over the field. Show that there's a $d$ with $d = c * ...
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1answer
31 views

Linear subspace of K[X]?

I got the following task from my professor and I wanted to ask for advice from you. Task: $K$ is a field I shall prove this statement Prove that for every $v$ element of $K$ the set $I_v$ = {f ...
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0answers
50 views

Are the following maps linear?

A linear map $T:V\rightarrow W$ is a function satisfying: $T(v_1+v_2)=T(v_1)+T(v_2), \forall v_1,v_2\in V$ $T(\alpha\cdot v_1)=\alpha\cdot T(v_1), \forall \alpha \in \mathbb F$ I am unsure if I ...
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2answers
42 views

Let $F$ be a linear operator such that $F^2 - F + I = 0$, show that $F$ is invertible and $F^{-1} = I - F$

I didn't understand this exercise. I tried working with $$F^2 - F + I = 0\implies (F-I)(F) + I =0$$ but I really don't understand how to prove $F$ is invertible neither find the inverse. Any hints? ...
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1answer
35 views

Properties of subspaces of a vector space

Let $U$ and $V$ be subspaces of the vector space $W$. Show that $$U+V=W \text{ and } U\cap V=\{0\}$$ holds if and only if for every vector $w$, there exists unique vectors $u\in U$ and $v\in V$ such ...
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0answers
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$V$ is $G$-irrep. over $\mathbb{C}$, submodules of $V \oplus V \oplus V$ given by imposing linear constraints. [on hold]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$. Let $W = V \oplus V \oplus V$. Show that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
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1answer
21 views

Linear Transformations of Functions

$\textbf{Problem}$ Define $f: \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) = mx + b$. $\textbf{a.}$ Show that $f$ is a linear transformation when $b = 0$. $\textbf{b.}$ Find a property of linear ...
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50 views

How to prove the following statements about a polynomial ring?

Edit: A user told me to split it into three questions, so just pay attention to the first statement here, I don't know if I should delete some questions here yet. I got the following task from my ...
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1answer
62 views

Linear algebra of state space representation won't be linear (superposition theorem)…

After answering a question about calculating the state space representation of a circuit with 3 sources in it (the circuit is there), I had a doubt - while checking, it became clear there is something ...
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1answer
20 views

Verifying if $F$ such that $F(1,0) = (2,5)$ and $F(0,1) = (3,4)$ is an automorphism

What I did: $$(x,y) = x(1,0) + y(0,1)\implies\\F(x,y) = xF(1,0) + yF(0,1)\implies\\F(x,y) = x(2,5) + y(3,4) = (2x+3y, 5x+4y)$$ I need to verify if $G = I + F$ is na automorphism. So: $$G = I + F = ...
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3answers
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Properties of vector spaces

Let $(V,\,\oplus,\,\odot)$ be a vector space. Let $u\in V$ and let $v$ be the additive inverse of $u$. $(i)$ Prove that $0\odot u = 0_{V}$. $(ii)$ Prove that $(-1)\odot u = v$. Note: $0_{V}$ ...
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1answer
31 views

Proving a property of vector spaces

Let $(V,\oplus,\odot)$ be a vector space. Let $u\in V$ and let $v$ be the additive inverse of $u$. Prove that if $w\in V$ is a vector such that $u\oplus w= w\oplus u = 0_{V}$, then $w=v$. Here I ...
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2answers
20 views

How to find the corresponding matrix of a dot product over a polynomial ring to a specific basis

Let $V= \mathbb R[x]_{\leq 2}$ be the vector-space of real polynomials with degree $\leq 2$. We define a dot product on the $V$ as follows: $$\left<f,g \right> = \int_{0}^1f(x)g(x)dx.$$ ...
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1answer
25 views

Prove the following Norm Inequality [on hold]

Show that $\forall x \in \mathbb{C}^n$ $$\|x\|_2 \leq \|x\|_1 \leq \sqrt n \|x\|_2$$
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24 views

Show Projection minimizes variance

Van der Vaart's Asymptotic Statistics, problem 11.2 Another idea of projection is based on minimizing variance instead of second moment. Show that $\text{Var}[T-S]$ is minimized over a linear space ...
2
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1answer
28 views

Generating Symmetric Matrix

Does anyone know how to generate random symmetric matrices whose minimum eigenvalue's multiplicity is at least 2? thanks
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2answers
41 views

Inequality $\sqrt[4]{x^TA^{-2}x}\sqrt{x^TAx}\leq 1$ for symmetric positive definite matrices

Assume that $x\in \mathbb{R}^{n}$ is a unit vector and $A$ is a symmetric positive definite matrix. Prove that $$\sqrt[4]{x^TA^{-2}x}\sqrt{x^TAx}\leq 1.$$ Progress Since A is spd, it is ...
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1answer
11 views

Tetrahedron in vector space: Finding a vector connecting two points

Edited to add: The tetrahedron is not necessarily a regular one. First off, the point $M$ is the centre of gravity for this tetrahedron. I have a base $\{e_1,e_2,e_3\} = ...
1
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1answer
14 views

Finding the zero vector of a vector space

Let $(V, \oplus, \odot)$ be a vector space with additive identity $0_{V}$. If $$(\exists z\in V)(\forall u\in V)\,\, \colon\quad u\oplus z = z\oplus u = u,$$ then $z=0_{V}$. Take $u\in V$, then the ...
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0answers
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Generic rank of tensors

Let the tensor product of the type $$ \underset{k=1} { \overset{m} \bigotimes } v_k$$ denote a simple tensor. As underlying fields, take $$ \underset{k=1} { \overset{m} \bigotimes } ...
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2answers
15 views

Rank of a matrix from a 5 X 7 matrix with a basis of 3 vectors

The question in my book is as follows: If the subspace of all solutions of Ax=0 has a basis consisting of thee vectors and if A is a 5 x 7 matrix, what is the rank of A? Now i thought because ...
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0answers
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Barrels of water, graphing an “acceptable” range…

So I have a bunch of barrels at a site that collect rain water. They’re all the same size, but the openings are scaled linearly from 100% of the lid size to nearly 0% on the “smallest” barrel. (but ...
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1answer
24 views

Given $\det(A)$ and $\det(B)$, is my calculation of $\det(-2B^T B A)$ correct?

Suppose $A$ and $B$ are $3 \times 3$ matrices with $\det(A) = -2$ and $\det(B) = -1$. What is the determinant of $C = -2 B^T B A$? I know that $$\det(A^T) = \det(A) \qquad \det(AB) = \det(A) ...
2
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1answer
26 views

Looking for example of a surjective homomorphism on $(\mathbb R,+)$ which is not an automorphism

Give example of a surjective function $f:\mathbb R \to \mathbb R$ such that $f(x+y)=f(x)+f(y) , \forall x,y \in \mathbb R$ but $f$ is not injective . I think I have to do something with basis of ...
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1answer
24 views

Covering of a vector space over a finite field

Let $k$ be a finite field and $V$ a finite-dimensional vector space over $k$. Let $d$ be the dimension of $V$ and $q$ the cardinal of $k$. Construct $q+1$ hyperplanes $V_1,\ldots,V_{q+1}$ ...
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1answer
17 views

Prove that if $C$ is anti hermitian matrix then $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $.

Suppose $C \in M_{n\times n}(\mathbb C)$ satisfies $C+C^* = 0$. Prove that $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $. Here is what I was able to show so far: We know that $C$ ...
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1answer
33 views

Find the number of surjective linear transformations

Let $V$ and $W$ be vector spaces over a finite field $F$ of order $q$ and $m=\dim(V)\geq \dim(W)=n$. Find the number of surjective linear transformations from $V$ to $W$. I know that if ...
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1answer
33 views

How to understand the concept behind the equation $\boldsymbol{Ax}=\boldsymbol{b}$

As is know to all, the equation $\boldsymbol{Ax}=\boldsymbol{b}$ can be understand as to find the linear combination coefficient of the column vector of the matrix $A$. At the same time, it can also ...
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Given a point on a plane, how do you determine the coordinates of the point in terms of the plane's vectors?

Suppose I have a plane $P$ and a line $L$ in $R^n$ $$ P(a, b) = \vec{p_0} + a * \vec{p_1} + b * \vec{p_2} $$ $$ L(c) = \vec{l_0} + c * \vec{l_1} $$ Say the line and the plane intersect at point ...
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2answers
74 views

Can we find the inverse for a vector

Can we inverse a vector like we do with matrices, and why ? I didn't see in any linear algebra course such a concept of vector inverse and I was wondering if there is any such thing and if not, why.
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2answers
92 views

What exactly are pseudovectors and pseudoscalars? And where could I read about them?

I can't find good information on the internet. In my mathematical physics class the definition of a vector was given as: That object with magnitude and direction which doesn't change under ...
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0answers
18 views

Local angle to world angle

I am using a digital gyroscope and I am getting very good results with it, only problem is the local angle does not match the world angle (seen by the world). Red = local X-axle Green = local ...
1
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1answer
30 views

Blockwise Symmetric Matrix Determinant

This question arises from another one of mine, but separate enough that I feel it deserves its own thread. Wikipedia says that $$det\begin{bmatrix}A&B\\B &A \end{bmatrix} = ...
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1answer
42 views

Need help with simple algebra equation

I would first like to impress that this is not a homework problem, but a personal one that I find myself unable to resolve, and that I am only an Algebra I student, and so I am unfamiliar with complex ...
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0answers
17 views

Finding scaling transformation to make vector elements add to 1 [solved]

I'd like to apply a scalar transformation to a vector in Rn such that the elements of the resulting vector sum to 1. I'm trying to do this to create weights for individual data points. Can someone ...
4
votes
3answers
217 views

All Two by Two Matrices Satisfy a Certain Property Problem

Show that if $A$, $B$ are $2 \times 2$ matrices over $\mathbb{R}$ then there exists a real number $\lambda$ so that $$ (AB-BA)^2 = \lambda I $$ I can do this problem using brute force (i.e. looking ...
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1answer
27 views

Courant minimax principle on block matrix

in going through some books about numerical mathematics I found the following exercise: Let $A,B \in \mathbb{R}^{n\times n}$ with $A$ symmetrical and rank($A$) = rank(B) = n. Define $M = ...
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43 views

$\text{Ker}A=\text{span}(u) \implies A=mat_C\left( u\wedge . \right)$

i found this equality and i wonder how can i find the right term $$\dfrac{1}{2}\left(\begin{matrix}0&1&1 \\ -1&0&1\\ -1&-1&0 ...
-3
votes
0answers
46 views

Prove the entries in the exponential of a matrix are finite real numbers [closed]

Let $A$ be an $n\times n$ matrix. Show that all entries of $\exp(A)$ are finite real numbers. I know that $\exp(A)=I+A+\frac{A^2}{2!}+\frac{A^3}{3!}+\cdots$ How do I show this? Any help is much ...
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0answers
29 views

orthogonal complement and direct sum proof

I don't know how to prove this statement. Someone can help me with is. Thanks!!!