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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51 views

Linear Algebra progression in our times

I know that Linear Algebra is relatively new branch of Mathematics. I wonder, Is there any significant progress with Linear Algebra in our time? Are there any major questions which haven't solved ...
3
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2answers
1k views

Determine the values of $k$ so that the following linear system has unique, infinite and no solutions.

Determine the values of $k$ so that the following linear system has a unique solution, infinite solutions and no solution. $2x + (k + 1)y + 2z = 3$ $2x + 3y + kz = 3$ $3x + 3y − 3z = 3$ I have ...
2
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3answers
823 views

$A$ is invertible matrix iff $Ax=0$ has the trivial solution only.

Why does the following statemnet true? $A$ is invertible matrix iff $Ax=0$ has only the trivial solution. My try: Let $x$ a solution of $Ax = 0$. Then, because $A$ is invertible there is ...
3
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1answer
73 views

Sum and difference of projections

I have been going through the Halmos' Finite dimensional vector space, where in section 42 I got struck at the theorems regarding the combination of projections. I can't understand how the conditions ...
1
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2answers
63 views

Do $N \times N$ identity matrices have N identical eigen values and infinite eigen vectors?

Suppose I have a $N \times N$ square matrix $A$. We know that for any matrix $A$, we have $N$ eigen values and corresponding eigen vectors. So a $2 \times 2$ matrix have two eigen vectors and ...
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0answers
50 views

Sylvester's determinant for matrix of grassman variables.

Assume one has two $n\times n$ matrices $A$ and $B$ containing only anticommuting grassmann variables. The task is to find the following product: $$\det(I-AB)\det(I-BA).$$ where $I$ is the identity ...
4
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2answers
558 views

Matrices A+B=AB implies A commutes with B

$A$ and $B$ are $n\times n$ matrices and $A+B=AB$. I have an interesting proof that this implies $A$ commutes with $B$, but the proof only works when $||B|| \lt 1$. I'm looking for a way to salvage ...
2
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2answers
58 views

Condition for a norm be absolute

Let $\|x\|_B\mathrel{\mathop:}=\sqrt{x^{t}Bx}$, where $B \in \mathbb{R}^{n\times n}$ is a symmetric and positive semidefinite matrix. If $\mid x\mid = (|x_1|,|x_2|,\ldots,|x_n|)$, I want to show that ...
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2answers
80 views

Double integral requiring orthogonal transformations and quadric forms

I have $\int_{-\infty}^\infty \int_{-\infty}^\infty exp(-x^T Ax) \;\mathrm{d}x_1 \; \mathrm{d}x_2$ $A = \left[ \begin{align} 3 && 2 \\ 2 && 3 \end{align} \right]$ Where $x^T = ...
1
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1answer
49 views

Eigen Values/Vectors of large scale data

Recently I have tried to learn the concepts of Eigen Values and Vectors and I have currently managed to calculate the Eigen values and vectors based off a 2x2 matrix based off calculating the ...
0
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1answer
69 views

I want some help of orthogonal vectors

$1.$ Let $u$ and $v$ be orthogonal vectors in $\mathbb{R}^n$ such that $\|u\|=2$ and $\|v\|=3.$ Find $\|2u+3v\|.$ I do it like $\|2u+3v\| < 2\|u\|+3\|v\| = 2.2 + 3.3 = 13$. $2.$ Let $u$ ...
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1answer
133 views

Change of basis of a gradient of a vector field

I'm working on my master thesis in computational quantum chemistry and need some math help. I have the gradient of the dipole moment of a 3-atom molecule in one basis, but need it in another. The ...
0
votes
5answers
141 views

Real $2\times 2$ matrix $X$ such that $X^2 + 2X= -5I$

Find a real $2\times 2$ matrix $X = \left(\begin{matrix} a& b\\ c & d\end{matrix}\right)$ such that $X^2 + 2X = -5I.$ With this question I'm kinda lost with the $2X$ part but a full ...
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1answer
38 views

Similarity A,B, and two other invertible matrices.

Well, if I have A similar to B. Is it necessary true for every two invertible matrices P,Q (dimension n) that $P^{-1}AP$ and $Q^{-1}BQ$ are similar? I guess it is. But I'm not sure. maybe a hint?
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2answers
64 views

Similarity AB,BA

Given:$ A_{n\times n} , B_{n\times n} .$ Is it necessary true that there is similarity between $AB$ and $BA$? I'm not quite sure how to check it is true for every non-singular M matrix. $M^{-1}ABM = ...
0
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1answer
45 views

Properties of a vector system.

Can someone help me with this example. An explanation will be very nice, even if you let me make the exercise. Is an example of an extra material I receive in my algebra class. Let $V = \{(x, y) ...
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3answers
34 views

Find a point $p_1$ on the line $l$ with distance d from the point $p_2$ on the same line

I have tried to find posts that are related to the question but they end up with the terms like ‘find a distance’. What I want is not to find the distance: I already have the distance, I want ...
0
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1answer
40 views

Parametric equations of manifold

I have am working for a linear algebra test and I realized that I don't know how to solve exercises with linear manifolds even the basic one. W : $ x+y-z+u=1 $ $ 2x+u=2 $ $ z -u=0 $ I don't ...
0
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1answer
238 views

Jacobian matrix and determinant - relation to orientation

$F$ is a function from $V$ to $V$ where $V$ is a $n$-dimensional vetor space and $p \in V$. In the article Jacobian determinant it says: "If the Jacobian determinant at $p$ is positive, then $F$ ...
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0answers
66 views

Orthogonal Matrices and similarity

Two questions: Is it necessary true that, Every two Invertible matrices with the same dimension are similar to each other. Every two row equivalence matrices with the same dimension are similar to ...
2
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2answers
102 views

Morphism of algebras

I would like to find some morphisms of $\mathbb C$-algebras $ \Phi : \mathcal{M}_3 ( \mathbb{C} ) \to \mathcal{M}_3 ( \mathbb{C} ) $ such that $$ \ \ \Phi( M ) \ = \ \Phi( J^{-1} M J ) \ = \ \Phi ( ...
2
votes
3answers
174 views

When to use $\times$ and $\otimes$

Im wondering when to use $\times$ and when to use $\otimes$. In some cases it seems very straightforward, for example $\times$ can be used when combining two elements into an n-tupel (for a product ...
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1answer
39 views

A Quick Set Notation Question

I've just been asked by a first semester linear algebra student to decide whether the following set is a subspace or not: $$ \left\{\{x_n\}_{n=1}^{\infty}\subset\mathbb{R}|\left(\forall ...
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2answers
228 views

Skanavi 2.003, Difference in answers, which is right?

In Skanavi book i have a exercise to simplify an equation $$\begin{align} ((\sqrt[4]{p} - \sqrt[4]{q})^{-2} + (\sqrt[4]{p} + \sqrt[4]{q})^{-2}) : \frac{\sqrt p + \sqrt q}{p-q} \end{align}$$ Solving: ...
2
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0answers
63 views

How to solve this system of inhomogeneous differential equations

In some past exam papers for the Maths course that I attend,I found this example and I would really appreciate if someone looked at my solution. It goes like this: Find general solution to $$ y_1' = ...
4
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1answer
96 views

Matrix invertability clarification and understanding

I am trying to set up some "mental models" for how to think about matrix invertability. I am currently studying linear algebra on a basic level and I would please like some explanations to the ...
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1answer
48 views

which of the following are true related to trace

Let, $A$ = ($a_i$$_j$) be a matrix of order $n$ and let $A^*$ denote the conjugate transpose of $A$. Which of the following statements are necessarily true ? $1$. If $A$ is invertible, then $tr$ ...
0
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1answer
54 views

Orthonormal basis of the subspace $2x-y+3z=0$

Given: subspace U: $2x-y+3z=0$ I should calculate an orthonormal basis for U. If the subspace is given as a span, I know how to approach. But how can I convert the equation above, to a span with ...
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0answers
36 views

Characterizing direct sums

Let $U,V$ be vector spaces. Let $T: U \to V$ be a linear map. The codimension of $T$ is defined to be $\mathrm{dim}(V) - \mathrm{dim}(\mathrm{im}(T))$. My questions are: (1) given the subspace ...
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1answer
144 views

Trace of the exterior power as a determinant

Let $A$ be a matrix. According to Wikipedia, $$tr(\wedge^k A) = \frac{1}{k!} \det \begin{pmatrix} tr (A) & k-1 & 0 & \cdots \\ tr (A^2) & tr (A) & k-2 & \cdots \\ \cdots & ...
2
votes
2answers
111 views

Inner product and linear transformation

Let V be an inner product space over $\mathbb{R}$ with inner product ⟨ , ⟩. Let $L:V\rightarrow\mathbb{R}$ be a linear transformation. Show that there is a $\vec{u}\in{V}$ such that $L(\vec{x}) = ...
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2answers
25 views

Diagonalisability with $\lambda = 2,x$

I have a $2 \times 2$ matrix, $Y(x)$, with $2$ eigenvalues: $\lambda = 2,x$ where $x\in \mathbb{R}$ Now $Y(x)$ is only diagonalisable if $\lambda_1,\lambda_2$ are distinct. Are $2,x$ distinct? I ...
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0answers
15 views

Finding normal components of a vector

If \begin{align} \notag A_{1}=c_{1}y_{1}^{2}y_{2}^{2}u(u+2t)(u+y_{2}^{2})\frac{\partial}{\partial x_{2}} \end{align} and \begin{align} \notag A=-c_{11}\frac{\partial}{\partial ...
0
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2answers
39 views

How can I know if sever functions are linearly independent?

Let $E$ be the collection of continuous functions with domain on [a,b]. Define $f+g(t)=f(t)+g(t),0(t)=0,(\lambda f)(t)=f(t)\lambda $, then $E$ is a vector space. I want to know how to determine if ...
12
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7answers
1k views

Why a subspace of a vector space is useful

I'm in a linear algebra class and am having a hard time wrapping my head around what subspaces of a vector space are useful for (among many other things!). My understanding of a vector space is that, ...
3
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1answer
86 views

Similarity transformation of a linear operator

I've seen in some books that given a differential operator $$\frac{d}{dx}$$ under a similarity transformation we get $$\frac{d}{dx}\rightarrow ...
0
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1answer
47 views

Solving system of 3 equations

How do I solve the following system? $$ \left\{ \begin{array}{} x_o = 4 - x_r \\ x_r = -2 - x_s \\ x_s = 2 - x_r \end{array} \right. $$ All the techniques i've found for solving 3-equation ...
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1answer
74 views

Does there exist $B$ for which $BB^T=I$?

My question is Does there exist a real matrix $B_{n\times m}$ with $m<n$ for which $BB^T=I_n$? Why do I need this? Suppose we are given a real matrix $Q_{m\times n}$ (again, with ...
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1answer
48 views

Is $f(X) = \| Y - XX^T \|_F$ convex given fixed $Y$?

In the scene of nonnegative matrix factorization, $f(X_1, X_2) = \| Y - X_1 X_2 \|_F$ is not convex, but both $f(X_1)$ given fixed $X_2$ and $f(X_2)$ given fixed $X_1$ are convex, enabling us to ...
2
votes
1answer
64 views

Matrix equivalence relation related to similarity and rank

The problem statement, all variables and given/known data. Let $X:=\{A ∈ \mathbb C^{n×n}:rank(A)=1\}$. Determine a representative for each equivalence class, for the equivalence relation "similarity" ...
3
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1answer
58 views

Vector spaces - $\min\{p\in\mathbb{N}|\text{ker}f^p=\text{ker}f^{p+1}\}=\min\{q\in\mathbb{N}|\text{im}f^q=\text{im}f^{q+1}\}$

$E$ is a $\mathbb{K}$ vector space, $f\in\mathcal{L}_\mathbb{K}(E)$. Let $p\in\mathbb{N}$ so that $\text{ker} f^p=\text{ker}f^{p+1}$ and $q\in\mathbb{N}$ so that $\text{im} f^q=\text{im}f^{q+1}$ ...
2
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1answer
54 views

Is a similarity map necessarily affine linear?

My text on fractal geometry introduces the following definition: A map $S: \mathbb R^n \to \mathbb R^n$ is called a similarity map if $$\exists c>0 \ \forall x,y \in \mathbb R^n: ...
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2answers
67 views

Prove that for $x,y\in \mathbb{R}^3,\lVert x \times y \rVert^2 = \lVert x \rVert^2 \lVert y \rVert^2 - \langle x,y \rangle^2$

Prove that for $x,y\in \mathbb{R}^3,\lVert x \times y \rVert^2 = \lVert x \rVert^2 \lVert y \rVert^2 - \langle x,y \rangle^2$ Use this identity to show that for $x,y \neq 0$, with the angle $\alpha$ ...
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0answers
34 views

Morphisms of algebras

Let's put : $ j = e^{i \dfrac{2 \pi}{3}} $, $ J = \begin{pmatrix} 1 & 0 & 0 \\ 0 & j & 0 \\ 0 & 0 & j^2 \end{pmatrix} $ and : $ \mathrm{Circ}_3 ( \mathbb{C} ) $ is the set of ...
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1answer
55 views

$A \in M_2(\mathbb{R}^2)$,$\langle x,y \rangle = x^TAy$ is an inner product iff $\alpha > 0, det(A) > 0$

Show that given $A=\left( \begin{array}{cc}\alpha &\beta\\\beta&\delta\\\end{array}\right) \in M_2(\mathbb{R}^2)$, $\langle x,y \rangle = x^TAy, (x,y \in \mathbb{R}^2)$, defines an inner ...
3
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2answers
94 views

Solve a system of linear equations

$\newcommand{\Sp}{\phantom{0}}$There is a system of linear equations: \begin{alignat*}{4} &x - &&y - 2&&z = &&1, \\ 2&x + 3&&y - &&z =-&&2. ...
4
votes
1answer
149 views

Bipartite graph matching like problem.

Let $A=\{a_1,a_2, ..., a_n \}$ and $B=\{b_1,...,b_m\}$ be finite sets. Also $A_1,...,A_k\subset A$ are covering of $A$ and $B_1,...,B_t\subset B$ are covering of $B$. Let $V$ be a set of pairs of ...
2
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1answer
64 views

Inner product definition, definite positive

I'm reading Hoffman and Kunze's linear algebra book and I'm a bit stuck with it's definition of inner product. One of the properties of the inner product is said to be that the product of any vector ...
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0answers
69 views

rank-one approximation for minimization of Frobenius norm out of kronecker products of unknown matrices

Let $A\in M_{pq}$ be a given matrix. We want to find matrices $X\in M_p$ and $Y\in M_q$ whose Kronecker product approximates $A$ as well as possible in the Frobenius norm; that is, we want to find ...
0
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1answer
23 views

System of linear equations with matrices

I am taking some online linear algebra classes. I would like to understand how to solve this system of linear equations. I used GEM to go to this step: $\begin{pmatrix}1 & 2 & 3 & 4 \\0 ...