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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Questions on color theory, expressed in linear algebra

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The ...
0
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0answers
15 views

Calculating the coefficients of a separable 2-qubit state

Given a separable 2-qubit state φ = φ0 ⊗ φ1 with φi= ai0|0> + ai1|1> φ thus can be written as φ = b00|00> + b01|01> + b10|10> + b11|11> with bij = a0ia1j. Now let some bij be given, i.e....
2
votes
0answers
36 views

Eigenvalue perturbation of singular matrix

Consider a Hermitian matrix $\mathbf{A_0} \in \mathbb{C}^{N \times N}$ with one singularity, i.e. its eigenvalues in increasing order are: \begin{equation} 0 < \lambda_2 \leq \lambda_3 \leq \cdots \...
0
votes
3answers
133 views

What is the number of subspaces of a particular dimension?

If we have vector space $V$ with dimension $n$ then how many subspaces of $V$ with dimension $m<n$ are there? In my opinion the answer should be the number of ways to choose $m$ linearly ...
0
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0answers
21 views

sending basissen

Lets say we have this $3\times3$ matrix: $$ \begin{bmatrix} 4 &−4 &12\\ 1& -1& 3\\ −1& 1 &−1 \end{bmatrix} $$ What is the algorithm to find a basis of $\Bbb R^3$ for which ...
-1
votes
0answers
12 views

Linear transformation and projection [on hold]

1 Suppose that W is a subspace of a finite-dimensional vector space V. (a) Prove that there exists a subspace W' and a function T:V→V such that T is a projection W along W'. (b) Give an example of a ...
2
votes
1answer
33 views

Row sum of inverse of a matrix

Let's say I have a matrix A, $$A= \begin{bmatrix} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \end{bmatrix} $$ All the elements of A are ...
0
votes
1answer
21 views

Is the condition number of unitary matrix always equal to 1?

I know that the 2-norm condition number $\kappa (\textbf U)={||\textbf U||_2}{||\textbf U^{-1}||_2}$ of a unitary matrix $\textbf U$ is always equal to 1. Is this true for all induced matrix norms, i....
2
votes
3answers
65 views

Baker-Campbell-Hausdorff/Zassenhaus formula to first order in one matrix

Is there a closed-form expression for the term of $e^{t(c \hat{X} + d \hat{Y})}$ that is first-order in $d$, where $t$, $c$, and $d$ are scalars and $\hat{X}$ and $\hat{Y}$ are finite-dimensional ...
1
vote
2answers
50 views

Compute $R^{2016}$ of a given counterclockwise rotation.

Write out the matrix $R$ of counterclockwise rotation by 30$^{\circ}$ in $\mathbb{R}^2$. Compute ${R}^{2016}$. Now this is an easy question to answer overall; 30 goes into 360 12 times and one twelfth ...
0
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0answers
26 views

problem solving in arithmetic

I've been given the following problem: The formula to find $Y$ is $Y=x_1+x_2+x_3+x_4-x_5$ The value of $Y$ is given as $100$. Now the question is: Is it possible to find without ambiguity $x_1$ ...
0
votes
1answer
48 views

$A^2$ is bounded $\implies$ $A$ is bounded?

Let $A_n$ be a sequence of $k \times k$ real matrices. Assume $A_n^2$ is bounded w.r.t some norm. Is $A_n$ also bounded? I was able to show this is true if $A_n$ are symmetric matrices (using SVD). ...
2
votes
0answers
26 views

If $AB = BA$ for $A,B \in \mathcal{L}(V,V)$, then $A$ and $B$ have these properties [duplicate]

There is a base such $A$ and $B$ are both upper triangular on these base, and if $A$ and $B$ are diagonalizable, then $A$ and $B$ are diagonalizable simultaneously. For the first I have no idea. To ...
1
vote
1answer
36 views

Algebraic number spaces

While studying about Vector spaces and subspaces I came across the following question:- $Q.$ Do $algebraic$ numbers form a subspace of the vector space $\Bbb R$? According to my knowledge of $...
0
votes
1answer
39 views

Coin toss related problem

What is the minimum number of times a fair coin needs to be tossed so that the probability of getting at least two heads is at least 0.96? Is there any shortcut way to calculate this?
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0answers
42 views

How to solve a function when given a graph? [on hold]

I'm not looking for the answer, but how to solve this myself. If there is a video I could be linked that would be very helpful. Thank you.
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vote
0answers
17 views

Derivative of quadratic form involving singularity

This might be a silly question, but i have been really curious about the following: Consider the following function seen thru a single variable, say $\alpha$: \begin{equation} f(\alpha) = \mathbf{x}^...
2
votes
1answer
96 views

Why must a function be independent of coordinates?

What is the motivation for why a function should be independent of coordinates? In the case of a general manifold I kind of get why, since one (usually) defines a function $f$ as a map from the ...
2
votes
2answers
61 views

Gradient and Hessian of function on matrix domain

Let $A \in R^{k \times p}$. Define $f(X) : R^{p \times k} \rightarrow R$ to be $f(X) = \log \det(XA + I_{p})$, where $I_{p}$ is a $p \times p$ identity matrix. I want to know what is the gradient and ...
0
votes
0answers
30 views

converting to row echelon form

Without swapping rows, transform the following augmented matrix into Row-Echelon form. $\left[\begin{array}{ccc|c}-1& 3& 2& -9\\ -2& 3& -2& -39\\ 1& -6& -5& 3\end{...
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vote
1answer
55 views

Implementing gradient descent based on formula

The gradient descent algorithm is given as : repeat { $$\displaystyle \theta_j := \theta_j - \frac{1}{m} \alpha \sum_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)}) x^{(i)}_j $$ } Given these values : <...
0
votes
0answers
25 views

Tensor product and dual of vector spaces

Consider $\mathbb{F}$ the algebraic closure of a finite field with characteristic $p>0$, and let $\mathbb{F}_q$ the unique subfield of $\mathbb{F}$ with $q=p^\alpha$ elements. So if we have $J$ ...
0
votes
1answer
21 views

Does a system of linear equation “equal ” to a matrix equation or it is just a trick to solve these equation?

Does a system of linear equation "equal" to a matrix form Ax=b or it is not equal(as if it just a "trick" to solve system of linear equation)?
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0answers
27 views

What would be a basis of $L^2(\Omega )$

Let $(\Omega ,\mathcal F,P)$ a probability space and $$L^2(\Omega )=\{random\ variable\ X\mid \mathbb E[X^2]<\infty \}$$ is a vector space. What would be a basis of $L^2(\Omega )$ ? I also know ...
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0answers
17 views

Associativity when solving matrix equation obtained from SVD

In this question, I asked how to solve a matrix equation $Ax=b$ using the Singular Value Decomposition (SVD) of $A$. The conclusion was rather trivial, given that $A$ is specified as square matrix. $$ ...
1
vote
1answer
57 views

Linear independence of standard basis vectors from Vandermonde style vectors

Is it true a statement that all $n$ dimensional vectors of the standard basis (e.g. $[1 \ 0 \ 0 \ ...]^T$, $[0 \ 1 \ 0 \ ...]^T$ etc ..) are linearly independent from the set of the $n-1$ vectors $...
1
vote
1answer
40 views

Doubly infinite matrices $A=(a_{i,j})_{i,j=\infty}^{\infty}$

Let $A=(a_{i,j})_{i,j=\infty}^{\infty}$, where $$ \|A\|:=\sum_{r=-\infty}^{\infty}\sup_{j}|a_{j,j+r}|<\infty. $$ I want to show that for all matrices $\|AB\|\leq\|A\|\|B\|$. I obverse that $$ (AB)...
4
votes
2answers
240 views

Union of a subspace and its orthogonal complement

The following statement seems to be true, but I am not sure: For any subspace $A$ of $\mathbb{R}^n$, there is a nonzero vector $\vec{x}$ such that $\vec{x}\in A\cup A^\bot$ and each entry of $x$ is ...
0
votes
0answers
24 views

How to solve an inverse problem $d=Ax_1 + Ax_2$

In the optimization problems, there is an operator, $A$, which transforms the model, $x$, to the data domain, $d$. Generally, we don't know the model and we are trying to find it according to the ...
2
votes
1answer
29 views

Representation of an invertible square matrix as a product of elementary matrices.

Suppose $A$ is an invertible $2 \times 2$ matrix. What is the smallest integer $n$ such that $A$ is a product of $n$ elementary matrices? My guess is that at most 4 elementary matrices are ...
0
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0answers
18 views

Explicit matrix representation of $\mathfrak{sl}_3$

Given a semisimple Lie algebra $\mathfrak{g}$ and a dominant integral weight $\lambda$ (and all the other necessary data), I want to be able to write down a matrix representation for $V(\lambda)$, the ...
0
votes
1answer
24 views

QR decomposition subcases

Is the full QR decomposition the most general, which includes the reduced QR, i.e, is it alright to always compute the full QR Decomposition for a given matrix blindly? What's the point of having two ...
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0answers
35 views

$V$ be a vector space , $T:V \to V$ be a linear operator , then is $(\ker(T) \cap R(T) ) \times R(T^2) \cong R(T)$? [duplicate]

Let $V$ be a vector space , $T:V \to V$ be a linear operator , then is it true that $(\ker(T) \cap R(T) ) \times R(T^2) \cong R(T)$ ? (note that the direct product is well-defined as both the spaces ...
0
votes
0answers
20 views

Compact notation for matrix multilplication.

I am confusing with some notation of matrix manipulation. Suppose write matrix $A$ as $A_{a}{}^b$, Then its transpose $A^T = A^b{}_a$, Is it right? I can see many textbook writhe its trace as $...
1
vote
1answer
45 views

Write out linear mapping of $L:\mathbb{R}^3 \to \mathbb{R}^3$

Write out the matrix $A$ of linear mapping of $L:\mathbb{R}^3 \to \mathbb{R}^3$ that projects $\mathbb{R}^3$ onto its linear subspace of columns with $x^2=x^3$ and parallel to the column $\left[0,1,-1 ...
3
votes
1answer
37 views

$V$ be a vector space , $T:V \to V$ be a linear operator , then is $\ker (T) \cap R(T) \cong R(T)/R(T^2) $?

Let $V$ be a vector space , $T:V \to V$ be a linear operator , then is it true that $\ker (T) \cap R(T) \cong R(T)/R(T^2) $ ( where $R(T)$ denotes the range of $T$ ) ? I know that the statement ...
1
vote
0answers
47 views

Null Space of a matrix $A$ [on hold]

For an arbitrary matrix, say $A$, Let $Q_1=\{x\mid Ax=0\ \&\ x^tx\ge0\}$, and $Q_2=\{b\mid b^tA^t A b=0\ \&\ b^tb\ge0\}$. Show that $Q_1=Q_2$.
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votes
1answer
16 views

Proper Use / Interpretation of homogeneous notation

I'm trying to understand the proper use or interpretation of homogeneous notation . I understand the concept of a linear transform $T$ such that $T(c_1\cdot\mathbf{x_1}+c_2\cdot\mathbf{x_2}) = c_1\...
0
votes
1answer
52 views

How to find eigenvectors given complex eigenvalues

I am given that: $\vec{x}' = A\vec{x}$, where $A=\begin{pmatrix} -3 & 0 & 2 \\ 1 & -1 & 0 \\ -2 & -1 & 0 \end{pmatrix}$ I want to find the general solution of this in terms ...
3
votes
0answers
38 views

Vectors arbitrarily close to subspaces

Let $X$ be a normed space and let $V$ be a nonzero subspace of $X$ which is not dense in $X$. I want to prove that for every $\epsilon>0$ there exists a unit vector $x\in X$ such that $0<d=\inf_{...
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1answer
41 views

How to simplify if given a one to one function? [on hold]

The one to one function is f(1)=3 and and I must simplify f(f^-1(5))+f(1)+f^-1(3). If I could just be given a video that teaches how to do similar problems that would be great. I couldn't find out ...
2
votes
1answer
26 views

Why is an operator composed with its adjoint positive and stricly positive when it's invertible?

Let $V$ be a (complex) finite vector space equiped with an inner product and $T$ an operator on V. We say $T$ is positive if: $$\langle T(v), v \rangle \geq 0$$ for all $v$ in $V$. We say $T$ is ...
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vote
1answer
30 views

Solving a recurrence relation of conditional probability functions

Suppose you have the recurrence relation for a probability function Q: $$Q(n_1,n_2|n) = Q(n_1-1,n_2|n-1)\frac{n_1-1}{n-1} + Q(n_1,n_2-1|n-1)\frac{n_2-1}{n-1}$$ where $n = n_1 + n_2$ and the ...
1
vote
1answer
54 views

How to prove that the vectors of the Krylov space of A are linearly independent if A is nonsingular.

$\mathbf{K}$ is a Krylov matrix. \begin{align} \mathbf{K}&= \left[ \begin{array}{ccccc} \mathbf{b} & \mathbf{A}\mathbf{b} & \mathbf{A}^2\mathbf{b} & \cdots & \mathbf{A}^{N-1}\...
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1answer
24 views

If $(A - \lambda I_n)$ is singular, then there exist $X \in M_n(\mathbb{C}), X \neq 0 $ such that $AX= \lambda X $

This question arise from Hoffman & Kunze - Linear Algebra, Sec. 8.5 Theorem 16. (page 313 on this edition). In the demonstratio it is given an argument about the existence of a root of the ...
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0answers
18 views

Finding Linearly independent solutions

Let $w<p<w'$ and $v\in \mathbb{F}_2^n$ be a random vector. How Can I find the sets of linearly independent vectors $v_i$ such that the linear combination of at most $p$ vectors in each set is $v$...
0
votes
1answer
25 views

A question on inner product space [duplicate]

I don't have much idea about inner product space. So, plz help me to understand this question; Let $A$ be a $n\times n$ matrix with real entries. Define $\langle x,y\rangle _A:=\langle Ax,Ay\...
2
votes
2answers
51 views

Given a symmetric matrix $A$, find $P$ such that $P^T A P$ is a diagonal matrix

Given $$A =\begin{pmatrix} 0 & 3 & 0 \\ 3 & 0 & 4 \\ 0 & 4 & 0\end{pmatrix}$$ find a matrix $P$ such that $P^T A P$ orthogonally diagonalizes $A$. Verify that $P^TAP$ is ...
1
vote
0answers
19 views

The norm of a simple linear functional

Suppose $X$ is a normed space, $x_0\in X$ and $M$ is a subspace of $X$. Suppose that $d=\inf_{m\in M} \|x_0-m\|>0$ and let $W=\text{Span } M\cup\{x_0\}$. Show that the linear functional $f\colon W\...
0
votes
0answers
20 views

Change eigenvalues of correlation matrix and transform into original basis

I use the Random Matrix Theory to filter out the information from the correlation matrix that is associated with noise - Marcenko Pastur band. That is straight forward. Then I follow Rosenow, Bernd, ...