Tagged Questions

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 ...

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3answers
453 views

Vandermonde matrix rank

Let ${\bf A} \in \mathbb{C}^{M\times N}$ be a Vandermonde matrix \begin{equation} \bf A = \begin{bmatrix}1&1&\cdots&1 \\ z_1&z_2&\cdots&z_N\\ ...
3
votes
1answer
194 views

Is a finite dimensional vector space always countable?

Given a vector space of finite dimension, can we always find an injective map to the natural numbers? z.
4
votes
1answer
271 views

Tricky problem on skew-symmetric matrices

Problem: If $S$ is a skew-symmetric matrix, show that $(I+S)(I-S)^{-1}$ is orthogonal. This appeared on a list of standard questions asked of Princeton graduate students. It has been a while since ...
2
votes
1answer
203 views

Diagonalization of a hermitian matrix?

It is said that any finite-dimensional Hermitian matrix can be diagonalized by a unitary matrix. But isn't a special unitary matrix sufficient? Am I making a mistake when I say that the phase that ...
1
vote
1answer
47 views

Real matrix and eigenvalues

A is real matrix from order nXn. We know that A gives : $<Av,v>=0$ for v vector in $R^n$. So what must exist? Every eigenvalue of A is real. A is not Invertible A is Hermitian A is not ...
2
votes
2answers
185 views

Prove that $\operatorname{Ann}(U \cap W) = \operatorname{Ann}(U) + \operatorname{Ann}(W)$ if $\dim V < \infty$

Prove that $\operatorname{Ann}(U \cap W) = \operatorname{Ann}(U) + \operatorname{Ann}(W)$ if $\dim V < \infty$ for $U$ and $W$ subspaces of $V$. Annihilator of U = Ann($U$) = $\{ \phi \in V^* | ...
0
votes
1answer
73 views

$\operatorname{Ann}(W)$ is naturally isomorphic to $(V /W)^*$

If $W$ is a subspace of $V$, show that $\operatorname{Ann}(W)$ is naturally isomorphic to $(V /W)^*$. Derive that $\operatorname{Ann}(\operatorname{Ann}(W)) = W$. Here Ann represents the ...
0
votes
1answer
74 views

$\operatorname{Ann}(U +W) = \operatorname{Ann}(U) \cap \operatorname{Ann}(W)$

Prove that if $U$ and $W$ are subspaces of $V$ , then $\operatorname{Ann}(U +W) = \operatorname{Ann}(U) \cap \operatorname{Ann}(W)$. I've tried this using the fact that $\operatorname{Ann}(\cup S_i) ...
2
votes
1answer
68 views

Set of functions as a vector space

Let $U$ be the set of all continuous functions $f: [a,b] \rightarrow \mathbb{R}$ such that $\int_a^b f(x) dx=1$. With the usual operations of pointwise addition and scalar multiplication, is $U$ a ...
1
vote
1answer
500 views

Need examples about injection (1-1) and surjection (onto) of composite functions

The task is that I have to come up with examples for the following 2 statements: 1/ If the composite $g o f$ is injective (one-to-one), then $f$ is one-to-one, but $g$ doesn't have to be. 2/ ...
1
vote
2answers
111 views

Variant of Cholesky Decomposition: solve $B^TB=A$ for general square matrix $A$

Choelsky Decomposition allows us to decompose a Hermitian Positive Definite Matrix $A$ as $A=LL^*$, and $L$ is guaranteed to be lower-triangular. My Question: If $A$ is non-Hermitian, but still ...
0
votes
1answer
66 views

circulant matrix question

could any one tell me what is translation? and if a $n\times n$ matrix $A$ commutes with all translation then must it be a circulant matrix? first please tell me the meaning of the question and then ...
0
votes
1answer
101 views

Need to expand $\nabla$ and $\Delta$ included term

In this equation, they used $\nabla$ and $\Delta$, I need to expand them to understand this equation. More to see in this article http://arxiv.org/abs/0802.3525 in equation (15) \begin{eqnarray} ...
2
votes
0answers
35 views

Continuity of Multilinear Maps [duplicate]

Possible Duplicate: Total Derivative and Multilinear Functions I was trying to figure out why a multilinear map on a finite-dimensional vector space is continuous...however I was stuck in ...
0
votes
2answers
75 views

insight on lemma on linear independence and spanning set

I've been staring at this lemma in my book on linear algebra for hours but I haven't managed to figure it out. Please help me understand. Lemma: Let $V$ be a finite-dimensional $K$-vectorspace with ...
6
votes
2answers
3k views

Find the roots of a polynomial using its companion matrix

I would like to find the roots of a polynomial using its companion matrix. The polynomial is ${p(x) = x^4-10x^2+9}$ The companion matrix $M$ is $M={\left[ \begin{array}{cccc} 0 & 0 & 0 ...
5
votes
2answers
315 views

A question about a method that shows $\mathbb{R} $ not finite dimensional.

Upon looking at methods that show $\mathbb{R}$ is not finite dimensional over $\mathbb{Q}$ I came across a method mentioned here by the user Bill Dubuque, he took a set of vectors of the form ...
1
vote
3answers
62 views

Computing the determinant of $\operatorname{id}+aa^t$

What is an easy way to see that $\det(\operatorname{id}_n+aa^t)=1+|a|^2$ for $a\in \mathbb{R}^n$ ?
0
votes
2answers
156 views

to check linear span of complex number

consider the set $\{ (1,0,-i),(1+i,1-i,1),(i,i,i)\}$ of three vectors from $\mathbb{C}^3$. which of the following is true? a) linear span of set is of dimension $1$ b) linear span of set is of ...
0
votes
1answer
130 views

The Transitive Property. Same relation or not?

Consider $\mathbf{u}, \mathbf{v}, \mathbf{w}$ vectors. If $\mathbf{u}$ and $\mathbf{v}$ are orthogonal, that is $\mathbf{u} \cdot \mathbf{v} = 0$, and the vectors $\mathbf{v}$ and $\mathbf{w}$ are ...
3
votes
1answer
113 views

Motivation for the double dual

I was reading on the double dual of a vector space $V$ recently. I was wondering what applications (within mathematics) there are for this concept and/or what was the motivation for the development of ...
1
vote
2answers
429 views

What does it mean to have a vector space of matrices?

Let $V$ be the vector space of all $m \times n$ matrices over some field $\Bbb F$. What does this intuitively mean?
7
votes
3answers
685 views

Suggestions for Studying for Real Analysis/Linear Algebra

I apologize if this question is inappropriate for this site, but I'm new here and am not entirely sure where to direct it. I've just begun a course on real analysis and linear algebra at my ...
3
votes
1answer
76 views

Is a topological subspace the same as a linear subspace in $\mathbb{R}^3$?

I am having a bit of trouble wrapping my head around the difference between linear and topological subspaces. For example, say we have the z-axis missing. Then $\mathbb{R}^3$ without it is not a ...
0
votes
1answer
263 views

Implications of zero row when row reducing matrix

Often when I am performing elementary row operations to row reduce an arbitrary $A_{m \times n}$ matrix, a row of 0's appears, $[0 \, \, 0 \, ... \, 0\, \, 0]$. I am uncertain, does this imply either ...
3
votes
1answer
162 views

Prove $\dim(W_1 +W_2) =\dim W_1+\dim W_2 - \dim W_1\cap W_2$

Show that if $W_1$ and $W_2$ are finite-dimensional subspaces of $V$ , then there exists a natural exact sequence $0 \rightarrow W_1 \cap W_2 \rightarrow W_1 \oplus W_2 \rightarrow W_1 +W_2 ...
0
votes
3answers
128 views

Simple linear algebra question

Let's say $A$ is an orthogonal $2\times2$ matrix over $\bf C$ and not diagonalizable over $\bf C$. Why then the determinant of $A$ must be $1$? I guess I'm missing something easy...
3
votes
3answers
111 views

Is there a counterexample to the claim that if $\mathbf y\cdot\mathbf y=1$ and $\mathbf x\cdot\mathbf y=c$ then $\mathbf x=c\mathbf y$?

Let $x,y$ be arbitrary vectors where $\mathbf{y} \cdot \mathbf{y} = 1$ and $c$ be a real valued scalar. If $\mathbf{x} \cdot \mathbf{y} = c = c (\mathbf{y} \cdot \mathbf{y} ) = (c \mathbf{y} ) \cdot ...
1
vote
2answers
92 views

Linear algebra eigenvalues and diagonalizable matrix

Let $A$ be an $n\times n$ matrix over $\mathbb{C}$. First I don't understand why $AA^*$ can be diagnosable over $\mathbb{C}$. And why $i+1$ can't be eigenvalue of $AA^*$? Hope question is clear ...
2
votes
1answer
287 views

Trace of a matrix times outer product

$\DeclareMathOperator{\trace}{tr}$Is there any relationship between $\trace(Sxx^T)$ and $x^TSx$? Is there a nice way to write the set of quadratic functions of the components of a vector $x$ given ...
6
votes
4answers
3k views

Rotating one 3-vector to another

I have written an algorithm for solving the following problem: Given two 3-vectors, say: $a,b$, find rotation of $a$ so that its orientation matches $b$. However, I am not sure if the following ...
0
votes
1answer
251 views

Columns of matrix $A_{m \times n}$ span $\mathbb{R}^{m}$

I am having trouble understanding the following theorem given in my textbook: Let $A$ be an $ m \times n $ coefficient matrix. Then the following statements are logically equivalent (That is, for ...
2
votes
5answers
129 views

Looking for proof of $\sum_1^n \alpha_i x^i = 0 \;\; \Longrightarrow \;\; \alpha_i = 0, \forall\;i$.

The assertion in the subject line is an abbreviated form of: $$\sum_{i=1}^n \alpha_i x^i = 0, \forall \; x \in {\mathbb R} \;\; \Longrightarrow \;\; \alpha_i = 0, \forall\;i \in \{1, \dots , n\}.$$ ...
0
votes
3answers
72 views

Linear algebra ( orthonormal basis)

$U$ is subspace for $\mathbb{R}^3$ with orthonormal basis $u_1,u_2$. Given $v\in \mathbb{R}^3,\;$ let $a_1=\langle v,u_1\rangle ,\;\; a_2=\langle v,u_2\rangle$ So it must be the case that: If ...
0
votes
4answers
172 views

linear algebra question

Consider the system \begin{align*} x + y + 2z &= 2 \\ 2x + 3y - z &= 5 \\ 3x + 4y + z &= b \end{align*} (a) For what values of $b$ does the system have a solution? Using this value of ...
1
vote
2answers
188 views

Finding the trace of a block matrix

Let $A$ be a $5 \times 5$ skew-symmetric matrix with entries in $\mathbb{R}$ and $B$ be the $5 \times 5$ symmetric matrix whose $(i, j)^{th}$ entry is the binomial coefficient $\binom{i}{j}$ for $1 ...
1
vote
0answers
20 views

conjugate space

Find the coordinates $\phi^\ast(\epsilon_{2}^\ast+\epsilon_{3}^\ast)$ in base $A^\ast$ where $A$ is $(3,4,5),(1,3,-2),(0,1,-2)$ and $M(\phi^\ast)_{st}^{st}=\begin{bmatrix} 1 & 3 & 2 ...
1
vote
2answers
109 views

Let $V$ be the inner product space consisting of linear polynomials

I am stuck with the following problem: Let $V$ be the inner product space consisting of linear polynomial, $p\colon[0,1]\rightarrow\mathbb R$ (i.e. $V$ consists of polynomials $p$ of the form ...
0
votes
2answers
233 views

Standard Basis of the Finite Field of Prime Numbers

A little info regarding this field: Addition and multiplication in $Z^n_p$ behave as usual but with the remainder taken upon division by $p$. Ex: $Z_3$ will only consist of the three integers ...
2
votes
1answer
49 views

Least-squares solution to an over-defined probem.

This problem arose in my stereo vision project. I have two matrix equations: $$\left( \begin{array}{ccc} x_1.w_1 \\ y_1.w_1\\ w_1 \end{array} \right) = \left( \begin{array}{ccc} A_{11} & ...
4
votes
1answer
76 views

The addition of linearly independent vectors.

If we say that $u$ and $v$ are linearly independent vectors in the vector space $V$ over some field $\Bbb F$, then $u+v$ is a linearly independent vector. Why is this? Isn't it possible for $u+v=0_v$ ...
0
votes
1answer
16 views

How to rearrange $(pgh_b)(s^2)-(pgh_t)(s^2)$ to $(pg)(s^2)(h_b-h_t)$

can someone please show me the steps to go from this $$ (pgh_b)(s^2)-(pgh_t)(s^2) $$ to this $$ (pg)(s^2)(h_b-h_t) $$ Thank you.
8
votes
2answers
283 views

Exponent of $GL(n,q)$.

Another exponent problem. $GL(n,q)$ is the group of invertible $n\times n$ matrices over the finite field $GF(q)$, where $q$ is a prime power. I am trying to figure out the exponent of this group. ...
0
votes
1answer
2k views

Converting parametric equation to implicit form

So I have the equation defined in homogeneous coordinates $[w; x, y]$ as $[1+t^2; 1-t^2, 2t]$ $$w = 1+t^2$$ $$x = 1-t^2$$ $$y = 2t$$ If I do $w+x-y$ I get $-2t+2$, so $t = -(w+x-y-2)/2$. I was then ...
3
votes
0answers
73 views

Jacobi's Rotation has two possibilities, why do they both result in same upper triangular magnitude norm?

The Jacobi's rotation is the complex Givens rotation (unitary similarity) that results in a zero for a specified element of a matrix. If the element is not adjacent to the diagonal, then there are ...
1
vote
2answers
268 views

How to workout the determinant of the matrix $D_n(\alpha, \beta, \gamma)$.

I am going through an example in my lecture notes. This is it: Let's introduce the matrix $D_n(\alpha, \beta, \gamma)$, which looks like this: $$\pmatrix{\beta & \gamma & 0 & 0 ...
3
votes
1answer
2k views

Need help with a linear equation with a free variable?

Is $a_1, a_2, a_3$ a linear combination of $b$? $a_1 = (1, -2, 0), a_2 = (0, 1, 2), a_3 = (5, -6, 8), b = (2, -1, 6)$ I used Gaussian elimination to get to. $$ \left[ \begin{array}{@{}ccc|c@{}} ...
2
votes
2answers
104 views

Functional in the dual space s.t $\psi$ restricted to a subspace equals $\phi$

Let $W$ be a subspace in a vector space $V$ [you may assume that dim $V$ is finite]. Prove that any $\phi$ in $W^*$ can be extended to a functional on $V$ , i.e. there exists $\psi$ in $V^*$ such that ...
0
votes
2answers
1k views

$V$ is isomorphic to $V^{\ast\ast}$, the double dual space of $V$.

Prove that for any vector space $V$ the map sending $v$ in $V$ to (evaluation at $v$) $E_v$ in $V^{**}$ such that $E_v(\phi) = \phi(v)$ for $\phi$ in $V^*$ , is injective. Derive from this that if ...
3
votes
2answers
272 views

Inverse Function Theorem/ Polynomial

I was thinking about this after I read about Jacobian conjecture. But I can't see what I did wrong? Maybe you can help me. Let $F: \mathbb{C}^n \to \mathbb{C}^n$ be of the form $F(x_1, \dots, x_n)= ...