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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2
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2answers
33 views

Dimension of $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$?

What is the dimension of an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$? Do I have to make distinct cases with as $p + q < n$ and equal to $n$? And if their ...
0
votes
3answers
52 views

I have difficulty understanding functions forming vector space.

I have knowledge of basic linear algebra, so I can understand the finite vector space as linear combinations of vectors of $R^n$. However, when it comes function as vector and functions form a ...
0
votes
1answer
36 views

Reflexive bilinear forms.

Let $V$ be a vector space and $B: V \times V \to \Bbb R$ be a bilinear form. Usually, I see books defining that if $B$ is symmetric, vectors ${\bf u},{\bf v} \in V$ are $B$-orthogonal if $B({\bf ...
0
votes
2answers
31 views

Linear Transformation one to one and onto?

Let $A = \left[ \begin{array}{ccc} 5 & -4 & 5 \\ 1 & -2 & -1 \\ -1 & 5 & 6 \end{array} \right].$ Is the linear transformation $T : \mathbb{R}^3 → \mathbb{R}^3$ defined by ...
0
votes
1answer
25 views

How to prove that the infinity norm of a matrix is the max of row sum?

I know how to prove that the 1-norm of a matrix is the max of the column sum, but not sure how to prove that the inf-norm is the max of the row sum. Any suggestion? Thanks
3
votes
1answer
29 views

Definition of complex conjugate in complex vector space

I am starting reading about Hodge theory and while reading the definition of abstract Hodge structure a very basic question came to my mind... What is the definition of the conjugate of a subspace of ...
1
vote
2answers
24 views

How to “compare” vectors?

I'm reading the definition of matrix norm in Golub & Van Loan and came across this "It is clear that the p-norm of matrix A is the p-norm of the largest vector obtained by applying A to a unit ...
0
votes
2answers
24 views

Interchange rows in a matrix without using interchange operation

I'm sure that it's already out there somewhere in the abyss that is page 37 on google, so I apologize. I haven't been able to find it. Given some arbitrary matrix, how can two rows be interchanged ...
0
votes
1answer
27 views

Find basis of subspaces

I don't know how to create basis of V1 and V2. If I want to prove M1^2=M1, do I need to find matrix representation of M1 first? Thanks!!!!!!
1
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0answers
23 views

Given a bilinear (or quadratic) form, how can you find the orthogonal of a vector space?

Let $V$ be a vector space over a field $F$ equipped with a symmetric bilinear form $B$. Let $W$ be a vector subspace of $V$. I know that we define the orthogonal complement $W^\bot$ to be ...
0
votes
2answers
15 views

consider if given vectors are elements of the span?

Consider the vectors u = (1,3,2) and v = (2,-1,1) in ℝ³. Determine whether or not (1,7,5) ∈ span(u,v) . Not really sure what to do, I was thinking of checking to see if u and v span ...
-1
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0answers
25 views

How to solve the vector differential equation? [on hold]

I'm new to this section, so I'm trying to solve vector differential equations, and I need some guidance. Could anybody give a step-by-step process for doing so, so that I could do some more problems ...
1
vote
1answer
12 views

Conditions of invertibility, linear transformations

Please, I need a hint. :) Let $T:\Bbb R^m\rightarrow \Bbb R^n$ and $ U:\Bbb R^n \rightarrow \Bbb R^m $ be linear transformations. What are the conditions that $m, n$ have to satisfy to $UT:\Bbb R^m ...
0
votes
1answer
27 views

Confused about Fourier series?

From linear algebra we know that if a set of vectors form a basis for a space, their is a unique linear combination of the basis to form any vector in that space. I'm assuming this extends to scalar ...
0
votes
1answer
23 views

Kernel of $q(x,y,z)=2x^2-4xy+2z^2-4xz+4yz$

I have some problems when calculating the kernel of the quadratic form $q(x,y,z)=2x^2-4xy+2z^2-4xz+4yz$: indeed, I get $Ker=\{(x,y,z)|x^2-y^2=0\}$, which results in a 2-dimensional kernel. Could you ...
0
votes
1answer
45 views

If two linear functionals are such that the kernel of one is contained in the kernel of the other, then they are proportional [duplicate]

Let $V$ be a vector space over $K$ and let $f,g \in V^*$ and satisfy $\ker f \subseteq \ker g$. Show there exist such $c \in K$ so that $c \cdot f =g$ How to approach this problem ?
1
vote
1answer
19 views

Matrix representation induced by quotient space

someone can help me with this question, I know how to solve ker(A) but I don't know how to develop matrix representation. Thanks!!!!!
1
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3answers
20 views

Find all linear operators such that $F^2 = F$ and $F(x,y) = (ax,bx+cy)$

I need to find all linear operators that match $F^2 = F$ and $F(x,y) = (ax,bx+cy)$ *where $F^2$ means $F$ composed with itself. So what I did: $F(x,y) = (ax,bx+cy)\implies F(F(x,y)) = ...
0
votes
1answer
19 views

Kernel of the quadratic form $q(x,y,z)=3x^2+6xy+10xz+2yz+3z^2$

Let $q:\mathbb{R^3} \to \mathbb{R}$ such that $$q(x,y,z)=3x^2+6xy+10xz+2yz+3z^2.$$ I have to determine rank, signature, kernel, and the canonic form of q (with its matrix). I have solved most of the ...
0
votes
0answers
9 views

How to Simplify/Rewrite this Expression into a Generalized Eigenvalue Problem - via Similarity perhaps?

I have the following optimization problem: \begin{eqnarray} min~b' y' Z (Z' \Omega Z)^{-1} Z' y b \end{eqnarray} such that $b'b=1$. The matrices are $Z \in R^{n \times k}$, matrices $y \in R^{n \times ...
1
vote
1answer
36 views

What is the derivative of (Ax)'

Let $f(x)=(Ax)^T$ where A is a matrix and x is a vector. How do you explain that $f'(x)=(Ax)^T$? Specifically, that $\frac{\partial}{\partial x} f(x) (y) = (Ay)^T$. I can't seem to do it rigorously. ...
0
votes
0answers
32 views

Lower and Upper Triangular Matrices

$A$ is an $n\times n$ matrix and $L$ is an $n \times n$ nonsingular lower triangular matrix. How can I prove that if $LA$ is lower triangular, then $A$ is lower triangular? How can I do the same for ...
1
vote
1answer
41 views

Help with finding basis of a vector space

Let $A = \left[\begin{array}{cc} 2 & -3 & 1 \\ 1 & -2 & 1 \\ 1 & -3 & 2 \\ \end{array}\right]$ and vector $u = \left[\begin{array}{cc} 2 \\ 1 \\ 1 \\ \end{array}\right]$ ...
0
votes
2answers
27 views

Sense of rotation. How would the rotation matrix look like for this “arbitrary” axis?

My first question is how do you define the sense of rotation about an arbitrary axis? Rotations are usually counterclockwise and when referring to rotation with respect to the $x$,$y$ or $z$ axis ...
-3
votes
1answer
32 views

What is true for rank of a $5\times5$ matrix [on hold]

Let $A$ be a $5\times 5$ matrix and let $B$ be obtained by changing one element of $A$. Let $r$ and $s$ be the ranks of $A$ and $B$, respectfully. Which of the following statements is/are correct: ...
1
vote
2answers
30 views

Why is a linear autonomous system asymptotically stable iff for all eigenvalues $\lambda$ of $A$, $Re(\lambda) < 0$

I'm trying to understand asymptotic stability of linear antonymous systems. I'm not sure if for the system $x' = Ax$, $x(t) = 0$ is the only fixed point that can be stable. In any case, I can ...
0
votes
1answer
19 views

How to find the base of Hom(U, V)

I know that Hom(U, V) is a vector space, which means it has a base. What is the way to find the base of it?
1
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0answers
20 views

Elementary Matrices, Replacing a row

Lets say I have a 4 x 4 Matrix and I want to replace one row of that matrix with a different row from the same matrix. I need a matrix that when multiplied to the original matrix achieves this. I ...
0
votes
1answer
20 views

Intersection of “positive” open half-spaces

Prove that the intersection of "positive" open half-spaces associated with any basis $x_1,x_2, \ldots, x_n$ of a finite dimensional vector space $V$ is non-empty. Recall that the "positive" open ...
2
votes
4answers
145 views

Definition of basis

There are something that I am not quite sure about the definition of basis. Let $V$ be a vector space over $K$, then the definition of basis says the vectors $v_1,...v_n$ form a basis of $V$ if they ...
1
vote
1answer
39 views

Vector Question… Stupid vector question.

Find a non-zero vector $\textbf u$ with terminal point $Q(-3,2,0)$. such that $\textbf u$ has the same direction as $\textbf v =$$ (3,1,-2)$ Since vectors are defined by their components and not ...
0
votes
1answer
26 views

Orthogonally diagonalizing a matrix

Can anybody explain how to orthogonally diagonalize the following matrix: $$ \begin{pmatrix} 9 & \sqrt10 \\ \sqrt10 & 0 \\ \end{pmatrix} $$ Am I correct in ...
0
votes
1answer
36 views

Find a basis for $\mathbb{R} ^5$ containing the given vectors

Find a basis of $\mathbb{R}^5$ that contains the vectors $(1,-1,1,-1,0)$, $(-1,-1,1,-1,0)$ , $(-1,1,1,-1,0)$. I think I need to find two more vectors so that the five vectors are all linearly ...
0
votes
1answer
22 views

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 $A^*A$. ...
1
vote
2answers
30 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& ...
-1
votes
0answers
24 views

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?
0
votes
1answer
15 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 ...
4
votes
4answers
181 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.
0
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0answers
28 views

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\\ ...
3
votes
2answers
32 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 ...
0
votes
1answer
25 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 ...
2
votes
1answer
132 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 ...
1
vote
0answers
54 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 ...
0
votes
1answer
37 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 $S_{n-1}$.
2
votes
1answer
27 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 ...
0
votes
1answer
31 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
votes
1answer
41 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
votes
0answers
15 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
votes
1answer
19 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 ...
1
vote
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 ...