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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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 ...
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25 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 ...
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36 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 ...
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Approximating Averaging : Signal processing

I read a paper reference at http://arxiv.org/pdf/1101.1764.pdf that if we average a set $V=\{V(t_0,\nu_0), V({t_1,\nu_1),..., V(t_n,\nu_n)}\}$; then we can approximate this average as: ...
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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 ...
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20 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 & ...
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1answer
18 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 ...
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27 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} ...
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16 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. $K$ ...
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1answer
21 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.
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30 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?
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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, $v_1$ ...
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14 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 ...
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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.
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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 ...
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3answers
48 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 ...
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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 ...
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27 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 ...
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1answer
24 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?
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1answer
20 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 ...
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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, ...
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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 ...
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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 ...
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1answer
24 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 $p(x) < 0$?
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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 ...
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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 ...
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3answers
43 views

Can $A$ be singular

$A^2 + A + I= 0$ Can $A$ be singular? Justify your answer. I do not know where to start.
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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 $x$). ...
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2answers
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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 ...
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1answer
19 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\\ ...
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1answer
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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 ...
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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} ...
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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 ...
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2answers
31 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) $P$ ...
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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 ...
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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 $x_1=x_2$ ...
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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 ...
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1answer
34 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 ...
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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 ...
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1answer
24 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 ...
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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 ...
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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 ...
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3answers
26 views

Elementary matrix proof

I am supposing that $E$ is the elementary matrix obtained from $I$ (the identity matrix), by adding $\mu$ times the $m$-th row to the $l$-th row for some $\mu \in \mathbb{R}$ and $1\leq l,m\leq n$ and ...
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2answers
29 views

Are all vector spaces also a subspace?

I am currently learning about vector spaces and have a slight confusion. So I know that a vector space is a set of objects that are defined by addition and multiplication by scalar, and also a list ...
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1answer
31 views

Is it true that $\|\text{diag($\pi$)} P\|_2 \leq 1$ for $P$ stochastic and $\pi P = \pi$.

$\| \cdot \|_2 $ is the matrix norm induced by $L_2$. $P$ is any given real square $n \times n$ non-negative matrix with rows summing to one, i.e. $P1 = 1$, where $1$ is the vector of ones. There is ...
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1answer
25 views

How to find out the linear transformation?

Is it linear transformation? Let the transformation be defined as $T:\mathbb{R}^3 \to \mathbb{R}$ $$T([x, y, z])=x^2-2y+3z$$ Well actually I have no idea how it works with an equation like ...
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1answer
12 views

Incorrect elementary row operation in an augmented coefficient matrix

When solving the matrix $$\left(\begin{array}{ccc|c} 1 & 1 & 1 & 4\\ 1 & 3 & 1 & 4\\ -1 & 2 & 3 & -2\end{array}\right)$$ I somehow made an error with the ...
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1answer
11 views

When is $\| X \| _\star = \| F X \| _2$ submultiplicative?

All matrices are real. By $\| \cdot \|_2$ denote the matrix norm induced by $L_2$. Assume $F$ is an invertible matrix. Consider the norm $\| X \| _\star = \| F X \| _2$. What is the condition on ...
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1answer
23 views

Linearisation of non-linear models

I'm asked to linearise the following model: $$y=\alpha xe^{\beta x}$$ I know I have to find an equation along the lines of $Y= A+BX$, but when I apply natural logarithms and use identities I still ...
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1answer
23 views

How to solve systems of linear equations of multiple variables (more than 3 to 100s)?

This was a question asked during an interview for programming job. And the bottom line was to write an alogrithm to solve such equations. As much as it numbed my neurons - it really provoked me. I had ...