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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Square root of matrix that is a square of skew-symmetric matrix

Let's suppose we have a matrix $A$ (dimension $3\times 3$) which is the square of some skew-symmetric matrix $S$ i.e. $A=S^2$. How to obtain from $A$ its skew-symmetric square root $S$?
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2answers
20 views

Matrices Eqvialence Relation

How can I prove that $A\mathcal{R}B$ is an equivalence relation if there exists an invertible matrix $C$ such that $B = CA$? I know there there is a reflexive, symmetric, and transitive steps. ...
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2answers
65 views

For what values of $a$ will $y=ax$ be a tangent to $x^2+y^2+20x-10y+100=0$

For what values of $a$ will $y=ax$ be a tangent to $x^2+y^2+20x-10y+100=0$ I tried to solve this question by differentiating and making it equal to $0$ and solving for $x$ and i got $-10$ as an ...
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2answers
31 views

From Cosine formula between two vectors to Schwarz inequality and Triangle inequality?

I'm studying linear algebra. I'm a beginner in this subject. The book says: Cosine formula of two vectors $$\frac{v \cdot w}{||v|| ||w||} = cos\theta $$ and it says, since $|cos\theta|$ never exceeds ...
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3answers
49 views

Prove that the product of two invertible matrices also invertible

I am working on a homework problem, but I am lacking some understanding. Here is the problem: Let $A$ and $B$ be invertible $n \times n$ matrices with $\det(A) = 3$ and $\det(B) = 4$. I know that ...
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0answers
21 views

linear algebra problem 12 [on hold]

In rotation in three dimensions, the square of length of any position vector does not change. Using the matrix representation of such transformation, show that three independent angles are sufficient ...
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0answers
35 views

Are the only 2x2 real matrices with complex-conjugate eigenvalues the rotation matrices?

If so, how can I see this fact? I'm wondering if it's something fundamental that I am overlooking. Thanks,
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28 views

Positivity of the last component of non negative least squares based on active set method

I have followed the instructions given in Lawson and Hanson book for non-negative least squares using active set method. I am having a trouble in justifying one of the statements they have made about ...
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0answers
20 views

Finding bases of subspaces of a linear map

So i'd like to find out a basis for the kernel and range of a linear operator over a polynomial field, but I'm having a little trouble with the kernel. Here's an example I made up consider ...
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2answers
148 views

Is there a fundamental theorem of algebra for matrices?

The fundamental theorem of algebra says we can do this ($z\in\mathbb{C}$ of course) $$\sum_{k=0}^n a_kz^k= a_n\prod_{k=1}^n (z-\omega_k)=0$$ for some set $\{\omega_k \in\mathbb{C}\}_{k=1,2,\ldots , ...
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3answers
25 views

Can p(1,2)= 1 and p(2,4)=3 be linear opperators?

This is a linear algebra question... So I know that the two conditions for linearity are additivity and homogeneity. Typically Ives seen examples where the ...
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1answer
25 views

A set of numbers whose product is a square

Let $p_{1}, . . . p_{10}$ be a set of ten prime numbers. We can construct $1023$ integer numbers by choosing any non empty subset of this set and multiplying all numbers in this subset. Find the ...
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1answer
20 views

Find the value of $k$ which will mean that these simultaneous equations have a consistent but not a unique solution.

The question is: Find the value of $k$ which will mean that these simultaneous equations have a consistent but not a unique solution. $$ 2x-2y+z = 10 $$ $$ 3x+y-3z = 18 $$ ...
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1answer
29 views

Matrix Notation! (Linear Algebra)

Suppose that we have a NxM matrix, where N=rows and M=columns. How could I write nicely a ...
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0answers
16 views

Matrix Derivative/Operation of Flat(A).dot(B)?

In calculating the gradient of a convolutional neural net by hand, I am running into a snag. In the middle of the net, going forward, there is a layer where I take an array $A$, flatten it into a ...
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2answers
39 views

Prove that 3d rotation is linear

In a 2d space, a transformation is linear if $f(v+w) = f(v) + f(w)$ and $f(kv) = k*f(v)$, and rotation preserves addition so it is linear. In a 3d space, similar rules apply: $(x, y, z) + (l, j, k) = ...
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1answer
35 views

Find a basis for a $2 \times 2$ matrix [on hold]

Let $W$ be the vector space of $2 \times 2$ matrices with real entries that add up to $0$. Find a basis for $W$ and use that basis to calculate $\dim(W)$.
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1answer
35 views

A decomposition of a finite-dimensional space

Suppose that $V$ is a finite-dimensional vector space over a field $k$ and suppose that $T: V \rightarrow V$ is a linear transformation. Set $V' = \cup_{n=1}^{\infty}\ker T^{n}$ and $V'' = ...
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2answers
27 views

Expressing a function as a linear combination

Hi so I know how to express a vector as a linear combination of another vector but now I've been given a function and it's thrown me a bit, just wondering if anyone can assist. The question is: In ...
2
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2answers
27 views

Angular momentum operators

Suppose we have angular momentum operators $L_1,L_2,L_3$ which satisfy $[L_1,L_2]=iL_3$, $[L_2,L_3]=iL_1$ and $[L_3,L_1]=iL_2$. We can show that the operator $L^2:=L_1^2+L_2^2+L_3^2$ commutes with ...
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1answer
38 views

Linear algebra for modern differential geometry( and other types of modern geometry, like analytic, complex and algebraic)

I wish to study real and complex analysis(for example, Pugh "Real Mathematical Analysis" and Cartan "Elementary theory of analytic functions of one and several complex variables") and modern ...
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0answers
21 views

Is this orthogonal distance a common pseudometric?

Define $d: V \times W \to \mathbb{R}$ such that $$d(v,w) = \sup_{z \perp w} \frac{\langle z, v \rangle}{\|v\|\|z\|}.$$ Is this a pseudometric that anyone has utilized in the literature? Does it have a ...
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7answers
1k views

Linear Algebra with functions

Basically my question is - How to check for linear independence between functions ?! Let the group $\mathcal{F}(\mathbb{R},\mathbb{R})$ Be a group of real valued fnctions. i.e ...
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0answers
24 views

Linear Transformation representation matrix.

I have some linear algebra question regarding let $\left(\vec{v_{1}},\vec{v_{2},}\vec{v_{3}}\right)$ be a base for$ \mathbb{R}^{3}$ and$ \left(\vec{w_{1}}\vec{w_{2}}\right)$ be a base for$ ...
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2answers
35 views

Determinant of map $p(x) \mapsto (Tp)(x)=a_n+a_{n-1}x+ \ldots +a_0x^n$

Let $V$ be the vector space of polynomial $\mathbb{R}$ of degree less than or equal to $n$. For $p(x)=a_0+a_1x+ \ldots +a_nx^n$ in $V$. Define a Linear Transformation $T:V \to V$ by ...
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1answer
24 views

How to denote dimensions

I am struggling with nomenclature. If I have matrix $M \in \mathbb{R}^2 \times \mathbb{R}^4$ it would be considered an element of an 8-dimensonal vector space. If I index $M$ by two indices $i$ and ...
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1answer
24 views

Does $A \succeq B A^{-1} B$ imply that $A \succeq B$?

Let $A,B$ be two symmetric matrices with equal dimensions. Suppose $A \succeq 0 $ (ie, PSD), $B \succeq 0$ and $$ A - B A^{-1} B \succeq 0.$$ Then is it true that $A-B \succeq 0 $?
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2answers
29 views

Let $u_1,u_2,\dots,u_n$ be a basis for $\mathbb{C}^n$ show that it is an orthonormal basis for some inner product.

Let $u_1,u_2,\dots,u_n$ be a basis for $\mathbb{C}^n$ show that it is an orthonormal basis with respect to some inner product. How would i go about doing this?
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14 views

Suppose $S,T\in\mathcal L \left({V}\right)$ are self-adjoint. Prove that $ST$ is self-adjoint if and only if $ST=TS$.

Suppose $S,T\in\mathcal L \left({V}\right)$ are self-adjoint. Prove that $ST$ is self-adjoint if and only if $ST=TS$. My attempt Since $ST$ is self-adjoint, $ST=(ST)^*=T^*S^*=TS$
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0answers
31 views

Help me with this linear algebra ques please! [on hold]

I still struggle with its concept. Please give me some help
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0answers
19 views

Finding a maximal orthogonal basis from a set of functions

I have a set of functions $\{f_1,\ldots,f_n\}$ with an associated inner product $\langle f_j,f_k\rangle=\int d^2z f_1^*f_2$ . The functions are not linearly dependent; i.e. the rank $r<n$, where ...
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2answers
30 views

Prove that $nullT^k=nullT$ and $rangeT^k=rangeT$

Supppose $T\in\mathcal L \left({V}\right)$ is normal. Prove that $nullT^k=nullT$ and $rangeT^k=rangeT$ For all positive integers $k$ My attempt I know that an operator on an inner product space is ...
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0answers
10 views

Exponential matrix using Laplace transform - reference request [on hold]

I am looking for a book that covers the concept of exponential matrix in detail using the Laplace transform plz
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1answer
47 views

Trouble to obtain eigenvectors of a matrix knowing its eigenvalues

The problem: Being given the matrix: $$ \begin{bmatrix} 0 & -1 & -1 \\ 1 & 2 & 1 \\ -1 & -1 & 0 \end{bmatrix}$$ and two of its eigenvalues ...
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1answer
24 views

How many ways are there to arrange the letters of word $ALGEBRA$ such that the relative order of the vowels and consonants doesn't change?

I did this question this way :- there are 4 consonants in the words (LGBR) and there are 7 letters in the word. $therefore$ number of in which consonants can be arranged in relative order will be ...
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1answer
23 views

Proving matrix similarity based on relationship between characteristic and minimal polynomials

Let $A,B$ be two $n \times n$ complex matrices which have the same minimal polynomial $M(t)$ and the same characteristic polynomial $P(t) = (t-\lambda_1)^{a_1}\cdots(t - \lambda_k)^{a_k},$ where ...
5
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1answer
57 views

Show that the number is irrational $\forall n$

I need to show that the number $\sqrt 2+ \sqrt[3]{3}+\sqrt[4]{4}+\sqrt[5]{5}+...+\sqrt[n]{n}$ is irrational for any n, and I don't have a clue about how I could show that. Thank you!
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1answer
19 views

proof on trace trick

I wanted to find the maximum likelihood estimator for $\mathbf{\Sigma}$ in the multivariate gaussian. I was anticipating the solution would be a bit involved and messy, if not 'brute-forced', but I ...
2
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2answers
50 views

Matrix that satisfies polynomial $x^{n}-1$ is diagonalizable

Let $A$ be an $n\times n$ matrix that satisfies the polynomial $p(x)=x^n-1$ (Update: $p$ is not necessarily the characteristic polynomial), that is $A^n-I=0$. I am trying to prove that $A$ is ...
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0answers
16 views

Point in $U$ closest to $(2,0,1,6)$

Let $U$ be the intersection of $x_1+x_2+x_3+x_4=0$ and $x_1+2x_2+3x_3+4x_4=0$ in $\mathbb{R}^4$. The basis of $U$ is $((1,-2,1,0), (2,-3,0,1))$ and the basis of the orthogonal space $U^\perp$ is ...
2
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1answer
41 views

Show that determinant is equal to determinant of each variable

Show that $$\begin{vmatrix} na_{1}+b_{1} & na_{2}+b_{2} & na_{3}+b_{3}\\ nb_{1}+c_{1} & nb_{2}+c_{2} & nb_{3}+c_{3}\\ nc_{1}+a_{1} & nc_{2}+a_{2} & nc_{3}+a_{3}\\ \notag ...
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2answers
49 views

Prove that there is a real linear polynomial $f(x)$

Let $(V,<,>)$ be a real inner product space and $T \in \mathcal{L}(V,V)$ a normal operator. Assume that the minimal polynomial of $T$ is a real irreducible quadratic. Prove that there is a real ...
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1answer
18 views

Uniqueness of Dual basis

I have read that the dual basis for the dual space of a vector space is unique. What does it mean in this sense to be unique? That it is the only basis possible or the only basis of this form?
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1answer
18 views

How to find basis of following space in the problem

Suppose a =(3,2,1,0) b=(1,1,0,0) c=(0,0,1,0) d=(3,2,0,2) e=(2,2,0,1) f=(1,1,0,1) are vectors in R_4 and the subspace of R4 spanned by a,b,and c is denoted by V, and subspace of R4 spanned by d,e and ...
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0answers
23 views

Bases of a Vector Space made up of union of two vector spaces [on hold]

If V and W are two distinct vector spaces and Y is a vector space which is a union of V and W. Then how to compute basis of W
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1answer
13 views

Generate a random binary full-rank rectangle matrix that is a basis of a subspace

Disclaimer: I think of vectors as row vectors. I have a full-rank $m \times n$ ($m < n$) binary matrix $B$ which is a basis of $m$-dimensional subspace $V \subset\mathbb F_2^n$ (i.e. subspace $V$ ...
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1answer
26 views

How to find matrix $A$ from the relation: $A\times (A^TA)^{-1}\times A^T = B$

Kindly help me in the following: I have two Matrices, $A$ of size $(n\times m)$; and $B$ of size $(n\times n)$, where $n>m$. $A$ is unknown, but $B$ is known. $(A^TA)$ is invertible $B$ is ...
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1answer
28 views

Practical use of matrix right inverse

Consider a matrix $A \in \mathcal{R}^{m \times n}$ In the case when $rank(A) = n$ then this implies the existence of a left inverse: $A_l^{-1} = (A^\top A)^{-1}A^\top$ such that $A_l^{-1}A = ...
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0answers
18 views

Exponentiation of Gell-Mann Matrices

The Pauli Matrices satisfy the relation $e^{ia(n\cdot\sigma)}=I\cos(a)+ ia(n\cdot\sigma)\sin(a) $. Can a similar equation be derived for the Gell-Mann matrices? The main feature of the Pauli ...
2
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0answers
71 views

Water filling problem in Blocks - Algebra Question

Consider a rectangular plot comprising $n\times m$ square cells on which $nm$ cement blocks of various heights are stored, one block per cell. The base of each block covers one cell completely, and ...