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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A connection between a matrix norm and a related matrix's largest eigen-value

I have been asked to prove that for $A\in M_n(\mathbb{C})$, with $||A||:=\sup_{x\in\mathbb{C}^n,|x|=1}|Ax|$, $$||A||=\sqrt{\lambda}$$ where $\lambda$ is the eigen value of largest modulus of $A^*A$. ...
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19 views

Similarity in two 2x2 Matrices and finding the S in A=SBS-1

I am doing something wrong here and I am not sure what. The object of the exercise is to find the S for similar matrices $A$ and $B$. $A=SBS^{-1}$ with $B=\begin{pmatrix}4& 1\\1& ...
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Im and Ker of matrix

If Ker are the solutions of the homogenous system and Im are for the $A^t$ matrix what is the point of defining them ?What is their purpose?
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10 views

When does the Singular Value Decomposition fail?

Does the singular value decomposition ever not work? The statement of the associated theorem, here from wikipedia: http://en.wikipedia.org/wiki/Singular_value_decomposition#Statement_of_the_theorem is ...
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38 views

Complex square matrices. Difficult proof.

$det(I+A\cdot\bar{A}) \ge 0$ Is it possible to prove the inequality is true for all complex square matrices $A$ where $I$ is the identity matrix and $\bar{A}$ is the complex conjugated matrix.
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Linear transformation of vector

I have computer graphics class and i had something like that on lecture: $$ \begin{bmatrix} \overrightarrow{b1} & \overrightarrow{b2} & \overrightarrow{b3} \end{bmatrix} \begin{bmatrix} c1\\ ...
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2answers
21 views

Rank of $ T_1T_2$

For n$\ne $ m let $ T_1 :R^n \to R^m $ and $ T_2:R^m\to R^n $ be linear transformations s.t $ T_1T_2 $ is bijective. Find rank of $ T_1$ and$ T_2$. I tried by fact that bcoz $ T_1T_2 $ is bijective ...
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21 views

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 ...
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19 views

Let $x, y \in \mathbb R^3$ . If $v$ is orthogonal to $x$ and $y$, then $v$ is a scalar multiple of $x \times y$.

Let $x, y \in \mathbb R^3$. If $v$ is orthogonal to $x$ and $y$, then $v$ is a scalar multiple of $x \times y$. How do I start to prove this? Please give me a hint.
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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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21 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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$V^{\oplus3}$, 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
30 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 ...
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38 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 ...
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13 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
17 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
30 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
16 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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30 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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40 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
41 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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22 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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$V$ 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
20 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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43 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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19 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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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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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
27 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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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
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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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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 ...
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1answer
25 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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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\} = ...
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1answer
13 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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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
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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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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
20 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) ...
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1answer
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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
23 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
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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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Do those two expressions have the same eigenvalues?

I am encountering the following eigenproblem \begin{eqnarray} \text{min} ~ \epsilon' Z' \Omega^{-1} Z \epsilon, \end{eqnarray} where $\epsilon$ is N by one, Z is N by K, $\Omega$ is K by K and real, ...
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29 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
32 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
71 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
85 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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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 ...