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 ...

learn more… | top users | synonyms (1)

0
votes
0answers
12 views

Matrix exponential using Cayley Hamilton

Matrix Exponential using the Cayley-Hamilton theorem This link provides a way but for a particualr simple looking matrix. Is there a general way for computing matrix exponential?
-1
votes
0answers
10 views

If you add row $1$ of $A$ to row $2$ to get $B$, how do you find ${ B }^{ -1 }$ from ${ A}^{ -1 }$?

If you add row $1$ of $A$ to row $2$ to get $B$, how do you find ${ B }^{ -1 }$ from ${ A}^{ -1 }$? Notice the order. The inverse of $B=\begin{bmatrix} 1 & 0 \\ 1 & 1 ...
0
votes
0answers
4 views

Prove that $J_n(0)$ and $(J_n(0))^t$ are similiar

Prove that $J_n(0)$ and $(J_n(0))^t$ are similar ($J_n(0)$ is a $nxn$ Jordanian block which belongs to the eigenvalue $0$). Use your answer and Jordanian form to prove that every matrix $A \in ...
0
votes
1answer
17 views

How to determine if (1,0,1,1), (1,1,0,1) , (0,1,1,1) spans $R^4$?

I set up a system where $a(1,0,1,1) + b(1,1,0,1) + c(0,1,1,1) = (1,1,1,1)$ (the standard basis of R4) then i found that $a + b = 1$ $b + c = 1$ $a + b + c = 1$ which implies that $a = c = 0,$ and ...
-1
votes
0answers
11 views

Confusion of intersection of two 2-d planes in 4-d

I've only just started a linear algebra course at my uni and I'm wondering if it is intuitive to say that two 2-d planes can't intersect in 4-d in such a way that they produce a 3-dimensional solution ...
0
votes
4answers
40 views

Prove that a product of two complex numbers has zero imaginary part

This is my homework, which reads as follows: Let $z_1, z_2$ be complex numbers. Prove that when $z_1z_2 \neq -1$ and $|z_1| = |z_2| = 1$, then the imaginary part of $$ \frac{z_1 + z_2}{1 + z_1z_2} $$ ...
2
votes
0answers
8 views

How to find the appropriate weights to maximize the third coordinate while the first two are zeros

Let's assume, that $v_1, ..., v_n \in \mathbb{R}^3 $ and $ \lambda_1, ..., \lambda_n \in [0, 1] $ The $ v_1, ..., v_n $ vectors are given. I have to find the appropriate weights ($ \lambda_1, ..., ...
2
votes
1answer
41 views

How can I solve for a , b , c , d?

Let's say I fix a list of two real numbers $\sigma = (\sigma_1, \sigma_2)$, and I want to show that there exists a real, entrywise-nonnegative matrix $A$ with $\sigma$ as its spectrum. How could I ...
0
votes
0answers
13 views

For these subsets $S$, are they subspace for the indicated vector space $V$

Q1. $V =P_5(R)$ and $S=\{p(x)\mid p(15)=0\}$. I think it is a subspace, but not 100% sure. I tried let $p_1(x)=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5$, such that $p_1(15)=0$ ...
2
votes
1answer
451 views

Solving a System of Linear Equations (k value for infinite, unique and no solutions)

$$x(k+2) + y(k-1) + z(k) = 2$$ $$y(k+2) + 2z(k) = 0$$ $$ z(k^2 + k -2) = k + 2$$ Determine the values of k for which the system has: Exactly one ...
1
vote
0answers
11 views

Finite field and its element with symbols [Sage / Python / …]

I have a finite field $T=GF(2^3)$, normal basis $(a, a^2, a^4)$ and polynomial $f$ from field $T$, which contains unknown variables / symbols. Is it possible to get vector with coordinates of f in ...
-2
votes
0answers
14 views

Addition of Kronecker Product Matrices

Summary: Is it possible to write $A_1 \otimes A_2 + B_1 \otimes B_2$ as some object which has nice properties again, preferably as a Kronecker product itself? Each of the matrices $A_i$, $B_i$ can ...
-1
votes
1answer
37 views

True or false? (linear algebra) [on hold]

If $u$ and $v$ are two solutions of $Ax$ = $0$, then any vector in $Span(u, v)$ is also a solution of $Ax = 0$. I have doubts with $span(u,v)$. I do not know if the same thing $span(u\cup v)$ or ...
0
votes
1answer
459 views

Change of (orthonormal) basis.

As I see it, the author says that $[Tv]_{e} = A[v]_{e}$ in the last paragraph. How do I see that ? I think I've jusitied the first entry in $[Tv]_{e} = A[v]_{e}$ as follows \begin{align*} \langle ...
0
votes
1answer
18 views

$Y$ coordinate of a point that lies on a line [on hold]

Given two points $A$ and $B$, for example $A(1,5),\,B(15,2)$, what is the $y$ coordinate of a point $C(x,y)$ lying on the straight line $AB$?
6
votes
0answers
40 views

In a finite dimensional inner product space with $T ∈ L(V)$, show that $\langle u,v\rangle = \langle T(u),T(v)\rangle$ implies $T$ is invertible.

Here is how I've tried to go about it, and I'm curious if it's true or if I'm way off base. T is invertible iff null$(T)=\{0\}$. Let $v∈V$ and suppose $T(v)=0$. If we can show that $v=0$, then $T$ is ...
-1
votes
0answers
19 views

How to explain polynomial coefficients by minimezed Error function?

We wish to predict ${\bf{t}}$ from an observed $\bf{x}$.We shall fit the data using a polynomial function of the form$$y({\bf{x}},{\bf{w}})=w_0+w_1x+w_2x^2+...+w_Mx^M=\sum_{j=0}^{M}w_jx^j$$ where $M$ ...
0
votes
0answers
16 views

bilinear form and positive definiteness

Let $B$ a symmetric bilinear form on an $n$ dimensional vector space $E$ with signature $(n-1,1)$. Then there exists a hyperplane $H$ in $E$ in which $B$ is positive definite. How to prove this? Is ...
3
votes
0answers
94 views

Problem involving subspaces and linear transformations

I'm asking for some opinions about my proof! $V$ and $W$ are vector spaces, and $T : V \rightarrow W$ is a linear transformation. $Z$ is a subspace of $W$, and $U$ is the set of all $\textbf{x} \in ...
7
votes
3answers
7k views

Prove rank $A^TA$ = rank $A$ for any $A_{m \times n}$

How can I prove rank $A^TA$ = rank $A$ for any $A_{m \times n}$? This is an exercise in my textbook associated with orthogonal projections and Gram–Schmidt process but I am unsure how they are ...
1
vote
2answers
13 views

Norm of diference of matrices of different rank

Suppose $A$ is a $n\times n$ matrix of rank $k$ that has Euclidean norm equal to $1$. Given $p<k$, and $\epsilon>0$, can we always find a norm one matrix $B$ of rank $p$ such that ...
3
votes
3answers
23 views

multiplication of finite sum (inner product space)

I am having difficulty to understand the first line of the proof of theorem 3.22 below. (taken from a linear analysis book) Why need to be different index, i.e. $m,n$ when multiplying the two sums? ...
-1
votes
0answers
15 views

finding eigenvalue and eigenbasis

How do I find the eigenvalue and eigenbasis of $$T(f(x))= f(4)x$$ Theres more to the equation but i just need to figure out how to apply $f(4)$ as a base in figuring out how to do the rest of the ...
1
vote
2answers
24 views

Prove a semi-positive operator $T$ is an isometry if and only if $T$ is the identity operator.

Prove a semi-positive operator $T$ is an isometry if and only if $T$ is the identity operator. I was thinking that semi-positive means if $T$ is self-adjoint ($T^{*}=T$) and $\langle T(u),u\rangle ...
0
votes
1answer
21 views

representation of a map with respect to 2 bases

From Heffron, p.231 Consider the two linear functions $h:$ ${R}^3$ $\longrightarrow$ $\mathcal{P}_2$ and ${g}: \mathcal{P}_2 → M_{2x2}$ $ \left( \begin{array}{ccc} a \\ b \\ c \end{array} ...
0
votes
0answers
12 views

Find the best approximate projection $w^*$ of $v^*$ onto $W$

Let $V=\mathbb{R}^3$, take $v^*=[1,1,1]^T$ and $W=span\{[1,0,0]^T,[0,1,1]^T\}$. Find the projection $w^*$ of $v^*$ onto $W$, that is, its the best approximation in the subspacce $W^*$. Let ...
3
votes
0answers
17 views

Compact-open topology on $\operatorname{Hom}_\mathbb{R}(V,W)$

Suppose $V$ and $W$ are finite-dimensional real vector spaces, and I give $\operatorname{Hom}_{\mathbb{R}}(V,W) \cong V^* \otimes W$ its usual vector space topology. Does this agree with the subspace ...
0
votes
1answer
18 views

Systems generators that are not linearly independent

Good evening, I would find sets that are generators of vector spaces, but they are not linearly independent, ie they are generating space but are not a basis for it. For example for these spaces: ...
2
votes
0answers
11 views

Where can I learn properties about spaces of linearly independent projectors?

I am interested in characterizing the space of all collections of $d^2$ linearly independent projectors on the Hilbert space $\mathbb{C}^d$. The linear independence I desire is in the vector space of ...
0
votes
0answers
22 views

Let $V = R^3$ and let $U$ be the subspace spanned by $A= \{(-3,-2,0),(4,-1,2)\}$. Is there a subspace $W$ of $V$ such that the following holds?

Let $V = \mathbb R^3$ and let $U$ be the subspace spanned by $A= \{(-3,-2,0),(4,-1,2)\}$. Is there a subspace $W$ of $V$ such that the following holds? $$W \nsubseteq U$$ ...
2
votes
1answer
531 views

How come that HSL can contain more information than RGB?

I have noticed weird thing when working with HSL - unlike RGB, it has some blind spots where certain value just does not matter. I'm sure we were taught about this when I had Linear algebra lectures ...
0
votes
1answer
10 views

Condition number of preconditioned system

Suppose we are solving an ill-conditioned system $Ax = b$, and we are trying to solve it using preconditioned technique. Given $\kappa (T)\approx \kappa(A)$, where $\kappa(A)$ is condition number of ...
1
vote
0answers
52 views

Cramer's rule doesn't work here?

I tried to solve the following system: $$A_2\cdot 2\mathrm{i}\sin( \beta a) = B_3\exp(- \alpha a)$$ $$\mathrm{i} \beta A_2 2\cos( \beta a) = - \alpha B_3\exp(- \alpha a)$$ Then I got $A_2=0 ...
3
votes
2answers
60 views

Efficiently solving many sets of linear equations without inversion or factorization

Suppose I have the normal set of linear equations $Ax = b$. If I can store and manipulate $A$ I have a variety of techniques available to me such as inversion, factorization, or an iterative method. ...
1
vote
4answers
39 views

How do I find orthonormal basis of $U$?

Let $U$ be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. How do I find orthonormal basis of $U$?
-3
votes
0answers
14 views

How do I set the vector $\vec{v} = (0, 0, 1, 0, 0)^T$ as $\vec{\vec{v}}=\vec{u}+\vec{w}$ with $\vec{u}\in U$ and $ \vec{w}\in U^\perp$? [on hold]

Let U be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. enter image description here
3
votes
2answers
58 views

Prove that $\|u - v\| \ge \|u\| - \|v\|$ for any $u, v \in \mathbb R^n$

Checking to see if the argument below works. Consider $\|u - v\|^2 = \|u\|^2 - 2(u \cdot v) + \|v\|^2$, by Cauchy-Schwarz: $\|u\|^2 - 2(u \cdot v) +\|v\|^2 \ge \|u\|^2 - 2(\|u\| \|v\|) + \|v\|^2 = ...
0
votes
1answer
35 views

Vector norm of $\mathbb R^n$, why is $p$-norm$\leq q$-norm if $p\geq q$?

Considering $$\infty\geq p\geq q\geq1$$ How can I show that the $p$-norm is smaller or equal to the $q$-norm? I can only show the case for $p=1, q=2$, but have no idea how to show others. Thank ...
0
votes
1answer
11 views

Generate a random binary full-rank rectangle matrix that is a basis of a subspace

Disclaimer: I think of vectors as row vectors. I have a full-rank $m \times n$ ($m < n$) binary matrix $B$ which is a basis of $m$-dimensional subspace $V \subset\mathbb F_2^n$ (i.e. subspace $V$ ...
-1
votes
2answers
17 views

How do I find a base of orthogonal complement $U^\perp$ of $U$ and determine the dimension of $U^\perp$? [on hold]

Let U be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. How do I find a base of orthogonal complement $U^\perp$ of $U$ and determine the dimension ...
4
votes
4answers
131 views

Finding a basis for the set of polynomials where f(1)=f(-1)=0

I have the vector space $V$ above that belongs to $\mathbb{F}$, and $V$ is the group of all polynomials that are of degree $3$. $W= \{ p \in V | p(1)=p(-1)=0\}$ 1.) Prove that W is a subspace of ...
1
vote
1answer
28 views

If $Ax = O$ has only one solutions, then the columns of A: ${v1, v2…,vn}$ span R?

I've been doing some excersices about inner product and I found something interesting but I don't know if my approach is correct at all. Supose that ${v_{1}, v_{2}, ..., v_{n}}$ is a base for a ...
1
vote
1answer
21 views

Show that the operator induced by $T$ on the quotient space $V/\operatorname{ker} (T-5I)$

A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^2(x-5)$. Show that the operator induced by $T$ on the quotient space ...
0
votes
2answers
24 views

The determinant of the transposing endomorphism

Let $K$ be a field and $f$ the endomorphism of $\mathcal M_n(K)$ that sends a matrix to its transpose. I want to determine the determinant of $f$. I know that since $f^2=id$ then $det(f)=1\ or \ -1 $ ...
0
votes
1answer
30 views

How do I determine $\min \left \| \vec{v}-\vec{u} \right \|_2$ for $\vec{u}\in U$?

Let $U=\lambda ((1, 0, 1, 0)^T,(1, 1, 0, 1)^T,(1, -1, 1; 0)^T$ is a subspace of $\mathbb{R}^4$. Determine for $\vec{v} = (1, 1, 1, 1)^T$ the vector $\vec{u}\in U$ minimal with $\left \| ...
0
votes
0answers
14 views

Can we exploit FFT for evaluating quadratic on gridded data with stationary covariance?

I would like to evaluate the quadratic $\mathbf{y}^{T}K^{-1}\mathbf{y}$ with the following assumptions: The entries of $\mathbf{y}$ are $y_i = f(\mathbf{x_i})$ which correspond to points on a ...
2
votes
0answers
116 views
+50

Spliting subspaces and finite fields

I'm sure that the following is true, but I can't prove it. Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a tower of finite fields and $A = \{\theta\in K: ...
2
votes
1answer
49 views

Basis for the intersection of two integer lattices

If $B_1$ and $B_2$ are the bases of two integer lattices $L_1$ and $L_2$, i.e. $L_1=\{B_1n:n\in\mathbb Z^d\}$ and $L_2=\{B_2n:n\in\mathbb Z^d\}$, is there an easy way to determine a basis for ...
1
vote
2answers
37 views

Is the set of periodic functions from $\mathbb{R}$ to $\mathbb{R}$ a subspace of $\mathbb{R}^\mathbb{R}$?

A function $f: \mathbb{R} \to \mathbb{R}$ is called periodic if there exists a positive number $p$ such that $f(x) = f(x + p)$ for all $x \in \mathbb{R}$. Is the set of periodic functions from ...
-4
votes
0answers
22 views

Suppose A and B are two matrices so that AB=0, which are true, if any, why? [on hold]

$ col(A)\subseteq null(B)$ $ null(A^TB^T) \subseteq null(A^T)$ $col(B^T) \perp col(A)$