# 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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### Prove that $\sqrt { 2 } +\sqrt { 3 } +\sqrt { 5 } +\sqrt { 7 } +\sqrt { 11 } +\sqrt { 13 } +\sqrt { 17 }$is irrational number? [duplicate]

I got this solution, but I didn't understand it. Assume that $b_1,b_2,b_3,\ldots, b_n$ are whole numbers (not zero). So, we have $b_1\sqrt{a_1}+b_2 \sqrt{a_2}+\ldots+b_n\sqrt{a_n}=0$. Prove it with ...
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### Matrix equivalence via orthogonal matrices

Let $A,B \in M_n(\mathbb{R})$. We say they are equivalent if there are $P,Q$ invertible such that $A=QBP$ (note this is weaker than similarity). Every matrix is equivalent to a diagonal matrix using ...
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### Gram matrix for a random variable vector space with inner product?

I am wondering if it is possible to construct a list of binary valued random variables, $\{\bf{X}_1,\bf{X}_2,\bf{X}_3\}$ and define a Gram-like matrix like \begin{bmatrix} \langle\bf{X}_1,\bf{X}_1\...
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### What can be said about $x^TAx$ is terms of $\|x\|$?

Let $A \in \mathbb{R}^{n \times n}$ and $x \in \mathbb{R}^n$ What can be said about $x^TAx$ is terms of $\|x\|$? I know that from C-S inequailty, $|x^TAx| \leq \|x\| \|Ax\|$, can I go further?
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### I'm having trouble understanding what this problem is asking me

This is the problem So my problem is that I dont know how to solve it... I have learned about system of inequalities and that kinda stuff, but I never got anything like this. I do not want anyone to ...
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### Operatornorm of $(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$

Determine the operatornorm of the mapping $I:(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$! Unfortunately I haven't many ideas for this task. I know that the definition of the ...
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### Whether a given algebra is the algebra of endomorphisms for a vector space.

Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms ...
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### How do I compute the area of this parallelogram

Given vectors $a,b$ and the ribs of parallelogram are $2a +3b = A$, $a-2b = B$. Also given $a \times b = (-1,2,2)$. Compute the surface of the parallelogram. I'm not sure where I saw but I think it ...
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### commutativity of log(I + A) and log( A−1) (matrix function)

I'm self-(re)learning linear algebra since the beginning of the summer, and i have a problem with the following exercice entitled additive logarithmic. If i'm right, we need to prove the ...
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### Showing properties of a function and its inverse image

I tried proving the following question but did not get too far. Let $\ f:A \to B$ be a function and $\ f^{-1}(Y)$ be the inverse image of $\ Y\subseteq B$ on $\ f$. Consider the following ...
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### Smallest possible value of the norm?

The vectors $\vec{u_1} = \begin{bmatrix} 1 \\ 1 \\ 1\\ 1 \end{bmatrix}$ and $\vec{u_2} = \begin{bmatrix} 1 \\ -1 \\ 1\\ -1 \end{bmatrix}$ are orthonormal in $\mathbb{R}^4$....
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### Application of Structure Theorem to Prove Simultaneous Diagonalizability and Group of Units of Cyclic Groups

I am reading these notes on Modules over PID. Exercise 67 (pg 24) asks to prove that: Problem. Let $A$ and $B$ be $n\times n$ matrices with complex entries. Then $A$ and $B$ are simultaneously ...
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### Trying to visualize and understand double dual space

Currently I am reading "Finite-dimensional vector spaces" by Paul Halmos. I would have a question regarding the theorem on page 25. It says: If $V$ is a finite-dimensional vector space, then ...
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### Why is this matrix symmetric?

There is an example in the Convex Optimization lecture notes, Boyd. He just said in the lecture that the matrix which is underlined in red color is symmetric! How can we claim that when there is no ...
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### Entanglement of 3-qubit states

Given a separable 3-qubit state φ = φ0 ⊗ φ1 ⊗ φ2 with φi= ai0|0> + ai1|1>, |0>, |1> being the computational base. φ thus can be written as φ = b000|000> + b001|001> + b010|010> + b011|011> + ...
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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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### 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 \$...
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### 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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### 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.