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 ...

learn more… | top users | synonyms

0
votes
0answers
8 views

linear functionals and inner product space

Let H be the inner product space of continuous real valued functions defined on [0,1] where ($\alpha$|$\beta$)=$\int_{0}^{1} \alpha(u)\beta(u)du$ Put K(s,t)=min{s,t}-st. Define T∈L(V,V) by ...
0
votes
1answer
11 views

Show tha the yz-plane is spanned by thes vector

Show that yz-plane w={(0,y,z):ybelongs to R} is spanned by (0,1,1) and (0,2,-1)
0
votes
0answers
5 views

Linear Transformations adn linear functionals

$F$=any field of characteristic 0. $V$=$F^3$, $W$=$F^4$ p∈L(V,W) given by p((x,y,z))=3x+4y+2z; q∈L(W,F) given by q((w,x,y,z))=2w+5x+7y+11z; T∈L(V,W) given by T((x,y,z))=(x,x+y,x+y+z,y+z) ...
1
vote
0answers
23 views

Show that $deg(f\cdot g)=n+m$

I started learning about rings and I was asked to proof some claims. I don't understand how I may prove the last one. I have proven that if $f$ and $g$ are polynomials over some ring of polynomials, ...
1
vote
0answers
23 views

$x \cdot x$ in inner product space is a quadratic form

Given an inner product space with some inner product $\cdot$ , how can I prove that $x \cdot x$ for any vector $x= (x_1,... x_n)$ is a quadratic form in $x_i$? I know how to recover an inner product ...
3
votes
0answers
12 views

Range of vectors that turn into eigenvectors after recursive multiplication by a matrix

Suppose $\mathbf{x}$ is a vector, and $\mathbf{A}$ is a square matrix. Which $\mathbf{x}$'s will satisfy the equation $\mathbf{A}^n\mathbf{x} = \lambda\mathbf{A}^{n-1}\mathbf{x}$, where $\lambda$ is ...
0
votes
0answers
23 views

problem in linear algebra [on hold]

Prove: If $A$ is invertible, then $AB^{-1}$ and $1+BA^{-1}$ are both invertible OR both not invertible
1
vote
2answers
29 views

Hilbert space: product and tensor product space

Let $H_1$ and $H_2$ be Hilbert spaces, then I would intuitively define the inner product on $H_1 \times H_2$ by $\langle (x_1,x_2),(y_1,y_2)\rangle = \langle x_1,y_1 \rangle + \langle x_2,y_2 ...
0
votes
2answers
42 views

Definition of sign

The following definition is in my notes with no explanation: $$\operatorname{sgn}(\sigma)=\begin{cases}1,&\text{if }\sigma(p)(x_1,\ldots,x_n)=p(x_1,\ldots,x_n)\\-1,&\text{if ...
2
votes
1answer
40 views

Develop good understanding of Linear Algebra

I am self studying Linear Algebra from book by Kenneth Hoffman and Ray Kunze, and currently I'm on 2nd Chapter of vector spaces, though the text is easy to follow, specially the exercises that follow ...
0
votes
0answers
11 views

Approximating Averaging : Signal processing

I read a paper reference at http://arxiv.org/pdf/1101.1764.pdf that if we average a set $V=\{V(t_0,\nu_0), V({t_1,\nu_1),..., V(t_n,\nu_n)}\}$, with $V(t_i,\nu_i)=e^{i\sigma(t_i,\nu_i)}$, then we ...
0
votes
0answers
11 views

Is it possible to have all the rows distinct(unique) in a matrix having only 0 or 1 , after removing exactly one column?

I have to solve the following programming problem Given a M*N binary matrix. Detect if it is possible to delete a column in a manner that after deleting that column, the rows of the matrix will be ...
0
votes
2answers
21 views

Find the matrix $A$ with this condition…

If $\theta \in\mathbb{R}\setminus\{k\pi, k\in\mathbb{Z}\}$ and $A\in M_{2\times 2}(\mathbb{C})$ such that $$A^{-1} \begin{pmatrix} \cos \theta & -\sin\theta \\ \sin \theta & ...
1
vote
1answer
19 views

Equivalent of solutions of IVP

Consider the IVP $y''-2y'+26y=0$, $y(0)=1$, $y'(0)=2$. From the characteristic equation $m^2-2m+26=0$, i found the roots as $m_1=1-5i$ and $m_2=1+5i$. Then when i use the basis solutions ...
1
vote
0answers
29 views

One question on Matrix Equation

Assume $\hat{M}_1, \hat{M}_2, \hat{T}_{11}, \hat{T}_{12}, \hat{T}_{21}, \hat{T}_{22}$ are $2\times 2$ matrix. And $a, b, A, B, C, D$ are all numbers, satisfying the following relation: \begin{align} ...
2
votes
0answers
18 views

Is there a nice representation for KKT conditions for matrix constraints?

I have a convex programming problem: $\min \left\lVert J - R \right\rVert _2$ $J,R$ are matrices. $J$ is given for the problem. One of the constraints is: $R = KQ$ Here, $R,K,Q$ are matrices. $K$ ...
1
vote
1answer
23 views

If A is a Hermitian matrix then SAS* is Hermitian

If $A$ is an $n\times n$ Hermitian matrix, and $S$ is an nxn matrix, then $SAS^*$ is also Hermitian. Why is this true? I have seen this claim made in several places but can't find a proof.
1
vote
2answers
32 views

Are three vector not in one plane mutually orthogonal, or linearly independent? [on hold]

Let $u, v, w$ be three points in $R^{3}$ not lying in any plane containing the origin. Are these three points linearly independent or mutually orthogonal?
0
votes
0answers
8 views

Can the Lanczos algorithm converge very fast by choosing initial guess smartly?

Suppose I have the two lowest eigenvectors $v_1$, $v_2$ of a matrix $M$. If slightly change $M$ to $M'$. Can I use $v_1$ or $v_2$ as an initial guess for $M'$? If so, which one should be used, $v_1$ ...
0
votes
0answers
25 views

Prove $NuclearNorm(W*U*S)\geq NuclearNorm(W*S)$

Suppose $W$, $S$ is two diagonal matrices of size $n*n$. $U$ is an orthogonal matrix. For $W$, the diagonal elements satisfies: $0\leq w_{1,1}\leq w_{2,2}\leq ...\leq w_{n,n}$, and for $S$, the ...
1
vote
1answer
19 views

Find a basis of a subset given an equation

$W = \{(x_{1}, x_{2}, x_{3})\in $R$^3: \frac{x_{1}}{3} = \frac{x_{2}}{4} = \frac{x_{3}}{2}\}$ Find a basis for $W$ I need help. I don't know how to do this.
0
votes
0answers
5 views

Prove $NuclearNorm(W*U*S)\geq NuclearNorm(W*S)$

Suppose $W$, $S$ is two diagonal matrices of size $n*n$. $U$ is an orthogonal matrix. For $W$, the diagonal elements satisfies: $0\leq w_{1,1}\leq w_{2,2}\leq ...\leq w_{n,n}$, and for $S$, the ...
0
votes
2answers
13 views

Determining whether sets of vectors form a basis

Is $\{(1,1,0,0),(0,0,1,1)\}$ a basis for the subspace of $\mathbb{R}^4$ consisting of all vectors of the form $(a,a+b,b,b)$ with $a,b\in \mathbb{R}$? Here is how I proceeded: First note that ...
1
vote
3answers
49 views

Linear Algebra: What do vector spaces represent?

I understand what a vector can represent. But I still don't understand what a vector space represents. I understand that you can add two vectors and that becomes a vector space. What else can you do ...
1
vote
1answer
17 views

Orthogonal transformation between vectors of the same norm

Suppose $V$ is a vector space over a field not of characteristic $2$, and is equipped with an inner product. I want to show that, given vectors $v$ and $w$, there is some orthogonal ...
1
vote
0answers
28 views

Compute a particular solution of AX = b

$A = \begin{bmatrix} 1 & 3 & 5 & 0 & 2 \\ 2&5&8&8&9 \\ 2&4&6&0&-1 \\ \end{bmatrix} $ Compute a particular ...
0
votes
1answer
25 views

Why is the laplacian matrix for a graph positive semidefinite?

Why is the laplacian matrix for a graph positive semidefinite? Can anyone provide an intuitive explanation and a proof?
0
votes
1answer
22 views

Find the equation of the ellipse with given foci and $a$

I have to find the equation of the elipse with foci: $$(-1,-1),(1,1)$$ and $a = 3$ I could do it using the definition of elipse, which I know how to work with. But I need to do it using translation ...
0
votes
0answers
9 views

Is there such a thing as a “continuum singular value decomposition”?

I have a question about expressing 2D functions as sums of separable functions. As a concrete example, consider the Gaussian circle function, ...
0
votes
1answer
16 views

Question about a subset not being a subspace in R^n

The question is: "Find an example of $S_{1}$ and $ S_{2}$ which are non-subspace subsets of $\mathbb{R}^3$ such that $S_{1}\cup S_{2}$ is a subspace of $\mathbb{R}^3$" I'm having trouble ...
2
votes
2answers
51 views

Proving ($\left|\left|Ax\right|\right| = \left|\left|x\right|\right|$, for all $x\in\mathbb{C}^n$) $\implies A$ is unitary

As the title states, I'm trying to prove that $\left|\left|Ax\right|\right| = \left|\left|x\right|\right|$ for all $x\in\mathbb{C}^n\implies$ $A$ is unitary, where $A$ is a square matrix. This is ...
1
vote
1answer
26 views

Vector subspace of polynomials

If I have a set of polynomials of degree at most $2$, such that $p(x) \geq 0$ for any real $x$. It isn't a vector subspace because I can multiply by a negative number such that $p(x) < 0$?
0
votes
0answers
9 views

Does the operation of addition on the subspaces of V have an additive identity? Which subspaces have additive inverses?

I was reading Linear Algebra Done Right. I came across the following question (Ch-1, Q12), for which I have solution , but I am having little confusion regarding it: Q12. (a) Does the operation of ...
0
votes
0answers
11 views

help with proof involving matrix derivations

So, Ive been trying to learn the research in a particular article, which can be read http://www.sciencedirect.com/science/article/pii/0024379580902219# Specifically lemma 2. So far, I have understood ...
0
votes
3answers
45 views

Can $A$ be singular

$A^2 + A + I= 0$ Can $A$ be singular? Justify your answer. I do not know where to start.
1
vote
0answers
29 views

Determining the set of a linear transformation of elements in a polyhedron

I have a set defined by linear inequalities of the form: $X = \{x : Ax \le b\}$. For any $x \in X$, I write $y = Gx$ where $G$ is a matrix (the dimension of $y$ is less than the dimension of $x$). ...
1
vote
2answers
41 views

Linear Algebra - Orthogonal problem (from exam can I appeal this?)

I have this problem : Let $A=\{v_1,v_2....,v_k\}$ in $R^n$ while $2 \leq k$. Prove if $A^\perp=(A-\{v_1\})^\perp$, then A is not linear independant. Please take a look at my solution since this is ...
1
vote
1answer
23 views

Relationship among $b_1$, $b_2$ and $b_3$ to have a solution

$$ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix} $$ If $b= \begin{bmatrix} b_1 \\ b_2\\ ...
0
votes
1answer
18 views

Show that $X_{\textrm{null space}} + X_{\textrm{particular}}$ is a solution of $AX = b$.

If $X_{\textrm{null space}}$ is a vector in $N(A)$ and $X_{\textrm{particular}}$ is a particular solution of $AX = b$, then show that $X_{\textrm{null space}} + X_{\textrm{particular}}$ is also a ...
1
vote
0answers
19 views

Commuting exponentials of non-commuting matrices

For two non-commuting matrices $A,B \in M(2,\mathbb{K})$, $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, can be shown that: $$ e^C=e^{A+B}=e^Ae^B=e^Be^A \iff \begin{cases} ...
2
votes
2answers
25 views

Linear Algebra: Symmetric matrices, diagonalization (help with proof)

I need a bit of help with an IFF proof, here it is: {Let X be a symmetric n × n-matrix. Show: $$X=Y^2$$ for some symmetric matrix Y iff X has only non-negative eigenvalues. } My thinking: This ...
0
votes
2answers
33 views

Equivalent quadratic form with 4 varibles

Consider two quadratic forms: $Q(x,y,z,w)=x^{2}+y^{2}+z^{2}+bw^{2}$ and $P(x,y,z,w)=x^{2}+y^{2}+czw$. For what type of values of $b$ & $c$ (real or complex or negative or positive or zero) $P$ ...
2
votes
1answer
17 views

Help with determining if a function is onto (surjective)

The question is to determine if the following function $T(x,y,z) = (y\sin x,z\cos y,xy)$ is onto. So far I have only learned of creating a coefficient matrix and checking if the determinant is $0$ to ...
0
votes
1answer
29 views

Find a $3\times3$ matrix whose reduced row echelon form has two leading ones and whose row space intersects its column space by the line $x_1=x_2=x_3$

I am confused on how a matrix can exist I tried doing something like this $$ \begin{bmatrix}1& 0& 1\\0& 1& 1\\0& 0& 0\end{bmatrix} $$ but this only intersects with $x_1=x_2$ ...
0
votes
1answer
24 views

Proof concerning basic solutions

Prove that every basic solution of $Ax=b$ (where $A$ is a matrix of rank $r$) is set by $r$ linearly independent columns of matrix $A$ (so it is $[A^{k_1}\dots A^{k_r}]\bar{x}=b$ where $A^{k_1},\dots ...
2
votes
1answer
36 views

Is $g(A)$ diagonalizable?

Let $A$ be an $n \times n$ diagonalizable matrix; let $g(x)$ be a polynomial. Is $g(A)$ diagonalizable? If not, what are the minimum hypothesis one needs to make so that it works (if any?) (As ...
0
votes
0answers
19 views

Find non degenerate linear programming problems

I have to find non degenerate linear programming problem in a canonical form such that: a) it has no solutions b) it has solutions, but but doesn't have an optimal solution A ...
0
votes
1answer
25 views

Find the Basis and dimension of orthogonal complement of W

$$U = \pmatrix{ a_1 & a_2\\ a_3 & a_4 } $$ $$ V = \pmatrix{ b_1 & b_2\\ b_3 & b_4 } $$ $$ \langle U,V\rangle = a_1b_1+a_2b_2+a_3b_3+a_4b_4 $$ $W= \{t(2, 0, 0, -1): t \in \Bbb R ...
1
vote
1answer
23 views

3D rotation around arbitrary axis

I have a 3D rotation matrix, R which is a combination of rotations around x-axis , y-axis and z-axis. I know how to calculate n(the arbitrary axis around which a point rotated about theta angle and ...
0
votes
1answer
22 views

Subspaces -vector spaces

Let V be a nonempty subset of R^n. Show that V is a subspace of R^n if and only if for all u,v ∈ V and c∈R,u+cv∈ V. Any1 can help with this ques?I don't really know how to show this.appreciate ur ...