# Tagged Questions

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 ...
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### 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....
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### 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: 0 < \lambda_2 \leq \lambda_3 \leq \cdots \...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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....
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### 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 ...
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### 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 ...
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### 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 $x_1$ ...
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### $A^2$ 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). ...
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### 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 ...
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### 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 : <...
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### 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 $J$ ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### $V$ 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 ...
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### $V$ 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 ...
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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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 $v$...