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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Finding rotation axis and angle to align two 3D vector bases

I have asked this question before and, while the accepted answer solved my problem back then, I am still interested in finding the rotation axis and angle. Let me rephrase the problem here: I would ...
2
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1answer
285 views

Prove that the determinant of $ A^{-1} = \frac{1}{det(A)} $- Linear Algebra

If I have a single matrix A that is non-singular, how can I prove the determinant of its inverse = $\frac{1}{\det(A)}$? Prove: $$ \det(\mathbf{A^{-1}}) = \frac{1}{\mathbf{\det(A)}} $$ I know that ...
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6answers
410 views

Lang's Linear Algebra: what's next?

I've completed the study of Lang's Linear Algebra ($3^\text{rd}$ edition). To put it simply, I have enjoyed the subject and I would like to know "what's next". In other words, I would like to know ...
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1answer
24 views

How to prove that $u\in V,\ 0\cdot u=\vec{0}$? [on hold]

Let $V$ be a vector space over a field $F$. How to prove: $$u\in V,\ 0\cdot u=\vec{0}$$
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1answer
18 views

Is Every Invariant Subspace the Kernel of an polynomial applied in the operator?

Let $V$ be a finite-dimensional $K$-vector space and let $T$ be a linear operator from $V$ to $V$. I already proved that for every polynomial $p(x) \in K[x]$, $\ker p(T)$ and $Im p(T)$ are ...
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1answer
21 views

Proving that the multiplicative identily $1$ is unique in $F$ [on hold]

Let $F$ be a field. How to prove: $$\exists\alpha\in F,\ \alpha\cdot\beta=\beta,\ \beta\in F \Rightarrow\alpha=1$$
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1answer
42 views

Find all reals $x$ such that $(x^3+2x)^\frac{1}{5}=(x^5-2x)^\frac{1}{3}$

Find all reals $x$ such that $$(x^3+2x)^\frac{1}{5}=(x^5-2x)^\frac{1}{3}$$ I reduced the question to find all positive $t$ such that $$(t+2)^3=t(t^2-2)^5$$ The solutions are $x=0$ , ...
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1answer
64 views

How can I find the common axis of 2 cones in space that have the same base radius but different heights?

How do I find the 3D vector describing the axis of 2 overlapping cones, like this: If I have only the following information: Coordinates of the common tip Coordinates of a point on the yellow ...
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0answers
10 views

How to solve this kind of difference equation?

How to find $v_k$, $k=0,1,2,\dots$ such that $$v_k + \sum_{n=1}^{k} \frac{\alpha^n}{n}v_{k-n} + \sum_{n=1}^{k}\frac{\beta^n}{n}v_{k+n} = 0,$$ where $\alpha,\beta \in \mathbb{C}$. ($v_i=0$ for ...
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2answers
76 views

Linear Transformations: Proving 1 dimensional subspace goes to 1 dimensional

I am having trouble understanding this whole question, and how to prove it. Let $F:\mathbb{R}^n\to\mathbb{R}^m$ be a linear transformation. Prove that if $L$ is a $1$-dimensional subspace of ...
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1answer
19 views

Proving full column rank of a matrix

Let $x$ be a $K\times 1$ vector of random variables satisfying that $E[xx']$ is nonsingular. For some given integers $M\geq 1$ and $L\leq K$, let $z_1,\ldots,z_M$ be $L\times 1$ column vectors ...
4
votes
1answer
50 views

Uniqueness of determinant

In Artin Algebra 2nd edition page 22, the author proved the uniqueness of determinant by saying that any matrix $A$ can be written in reduced row-echelon form $A'$: $A'=E_1\cdots E_kA$ where $E_i$ are ...
2
votes
1answer
25 views

Kernel and diagonalizability of endomorphism $f:\mathbb{R_2[x] \to R_2[x]}$ such that $f(p)=p(1)x^2-p(k),$ for $k \in \mathbb{R}$.

Problem: Let $f:\mathbb{R_2[x] \to R_2[x]}$ be the endomorphism on the space of polynomials of degree less or equal than two such that $$f(p)=p(1)x^2-p(k),$$ for $k \in \mathbb{R}$. I have to ...
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1answer
18 views

Why $(f\mapsto f(v_i)w)_{i,j}$ with $f\in V'$,$w\in W$ is a basis of $\mathscr{L}(V',W)$?

I'm trying understand the proof of the Proposition 3.1.2 (pg.5) of this document: http://www.win.tue.nl/~amc/ow/lba/lba3.pdf Suppose $V$ and $W$ are finite dimensional. If $(v_i)_i$ is a basis of ...
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1answer
56 views

What's the difference between these two definitions of polynomial function?

Definition 1: Given $a_n,...,a_1,a_0 \in \mathbb{R}$, a polynomial function is a function $p:\mathbb{R} \rightarrow\mathbb{R} $ such that $p(x)=a_nx^n+...+a_1x+a_0$ Definition 2: The function ...
2
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1answer
57 views

Eigenvalues of symmetric matrices are real without (!) complex numbers

Is there any proof of the fact that the eigenvalues of symmetric matrices (i.e. $A\in\mathbb{R}^{n\times n}$ with $A^t=A$) are real without the use of the concept of complex numbers?
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0answers
11 views

Dimension of sum of permutations of tensor products of vector spaces

Sorry for the mouthful of a title! Suppose I have two finite vector spaces $W,V$ with bases $\{w_1\dots w_p\}$ and $\{v_1\dots v_q\}$. Consider some subspace $S$ of $W\otimes V$ of dimension $m$ ...
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1answer
287 views

Orthogonal Complements and Subspaces Proof

I'm having a little difficulty understanding the proof for orthogonal complements. I kind of understand orthogonal complements, but I cannot seem to find a logic to this. I'm trying to follow along ...
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1answer
38 views

Linear Transformation from $\mathbb R^n$ to $\mathbb R^m$: image of $1$ dimensional subspace has dimension $1$ or $0$ [duplicate]

I am struggling to comprehend the question below. Especially the meaning of 'the image of $L$ under $F$'. Let $F : \mathbb R^n \to \mathbb R^m$ be a linear transformation. Prove that if $L$ is a ...
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1answer
20 views

Expressing a $SL_2(\mathbb{R})$ matrix as product of…

If $\begin{bmatrix} a&b \\ c&d \end{bmatrix}$ is some matrix in $SL_2(\mathbb{R})$, then how can we express it as a product of matrices of the following type: $$\begin{bmatrix} s&0 \\ ...
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0answers
14 views

Unit balls and the Schatten norms

I have a very naive question: Let $A$ and $B$ $n \times n$ (complex) matrices with operator norms $\|A\| \leq 1$ and $\|B\| \leq 1.$ Pick a $1 \leq p < \infty.$ Then with a constant $K_p$ ...
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1answer
39 views

Formula to calculate angle on a fan or semicircle

How do I calculate the angle shown in the picture given the height, width, and the arc deduction of $2$? I had applied the Right Triangles formula to calculate the hypotenuse: $h^2 = a^2 + ...
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1answer
31 views

Eigenvalues of a Product of two matrices A and B inside trace operator expressed in terms of any eigenvalue of A or B?

This question has been in asked in a few varieties here but not in this one. If we have a real, symmetric, positive-definite matrix $A$ and a real, symmetric, positive-definite matrix $B$ and we know ...
1
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0answers
30 views

Generic rank of tensors

Let the tensor product of the type $$ \underset{k=1} { \overset{m} \bigotimes } v_k$$ denote a simple tensor. As underlying fields, take $$ \underset{k=1} { \overset{m} \bigotimes } ...
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0answers
9 views

Differentiable vector space valued functions doesn't depend on basis?

Differentiable vector space valued functions. Let $V$ be a vector space over $\mathbb F^n$ ($\mathbb R$ or $\mathbb C$) and let $v_1, \ldots, v_n$ be a basis for $V$. Define the linear isomorphism ...
2
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1answer
18 views

Clarification of some doubts: working with the restriction of a quadratic form

Let $q:\mathbb{R^3}\to\mathbb{R}$ such that $$q(x,y,z)=2x^2+3y^2+4xy-2xz.$$ I have to determine rank and signature of $q$, and so far it should be fine: I got $\operatorname{rk}(q)=3$ and ...
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0answers
48 views

Why the space of all permutations of a vector (n!) is smaller than the space of all possible permutations of a sorting network?

Imagine you have a vector with 2048 entries. The total permutations are 2048! Now you have a sorting network let us say AKS, the total number of possible results with nlog(n) gates is $2^ {n log (n)}$ ...
2
votes
1answer
42 views

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
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1answer
19 views

Matrix associated with a bilinear form

We have $$b(v,w)=\begin{pmatrix} x_v& y_v& z_v \end{pmatrix} A \begin{pmatrix} x_w \\ y_w \\ z_w\\\end{pmatrix},$$ (where $A$ is the matrix associated with the bilinear form $b$ defined on ...
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2answers
24 views

Orthogonal complement $V^\bot$ of the vector space $V=\langle(1,0,2),(3,-1,0)\rangle$ and $V\cap V^\bot$

Consider the inner product defined by polarizing the quadratic form $$q(x,y,z)=x^2-z^2+4xy-2yz$$ on $\mathbb{R}^3$. Let $V=\langle(1,0,2),(3,-1,0)\rangle$. Could you show me how to find $V^\bot$ and ...
2
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1answer
12 views

Logic supporting column operations on matrices

In matrices, we justify row operations by drawing parallels with solving a system of equations i.e.: 1.Interchanging rows = Interchanging equations \ 2.Adding one multiple of a row to another = ...
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1answer
2k views

Understanding how to find a basis for the row space/column space of some matrix A.

I just need some verification on finding the basis for column spaces and row spaces. If I'm given a matrix A and asked to find a basis for the row space, is the following method correct? -Reduce to ...
1
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1answer
13 views

Visualizing cross product of points in 3-Space

If $p_0, p_1, p_2$ are three distinct points in space, then what does the cross product $$n = (p_0 - p_1) \times (p_0 - p_2)$$ mean geometrically? I'm having a little trouble visualizing this in ...
1
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1answer
13 views

Determining matrix M from $Mx_1 = b_1$ & $Mx_2 = b_2$, where $x_1, x_2, b_1, b_2$ vectors?

I have 4 vectors in the plane, $x_1$ and $x_2$, $b_1$ and $b_2$, and I'm told that there is a matrix $M$ such that $Mx_1 = b_1$ and $Mx_2 = b_2$. If I have a vector $x_3$, how do I determine $Mx_3$? ...
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2answers
33 views

Reference Request on a good Linear Algebra book [duplicate]

So I'm looking for a linear algebra book with a strong focus on proofs. It would be great if the book also uses concepts from regular abstract algebra like isomorphisms etc instead of dancing around ...
4
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0answers
34 views

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors?

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors? For example: $f=x^5+3x^4+x^3+4x^2+1$, and $g=x^5+3x^4+4x^3+3x+1$ Can we represent these ...
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1answer
21 views

which one is non singular matrix [on hold]

let A,B be n-square matrices such that $$BA+B^2 =I-BA^2$$ where I is the n-square identity matrix . which of the following is always true 1. A is non-singular 2.B is non singular 3. A+B is non ...
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0answers
21 views

Car-van ratio problem [on hold]

There are 2/5 as many vans as cars and 2/3 as many motorbikes in a parking lot. What is the ratio of vans to the total number of cars and motorbikes?
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1answer
12 views

Prove: If $\Gamma$ is a collection of subspaces that is totally ordered by set inclusions, then the union of all members of $\Gamma$ is a subspace.

I have been mulling this problem over in my mind for the last couple days and I am stuck. There must be some basic principal I am missing. Closure with respect to scalar multiplication is obvious. ...
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0answers
12 views

Is R(A) = ker(A^t)?, where R(A) is the space generated by the columns of A

I'm looking at this deduction of the normal equations that solve the linear least squares problem. It goes like this: R(A) is the space generated by the columns of A $\hat{X}$ is the solution of the ...
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1answer
27 views

the values of K in the following system [on hold]

I was assigned some homework for selfstudy but i cant make head nor tails of it. the assignment: for wich values of k in the following system does the system have no solutions one solution ...
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2answers
155 views

proving:Lamda is an eigenvalue of T if and only if $T-\lambda \mathrm{I}$ is not injective.

I'm trying to prove a theorem from "linear Algebra Done right" by Sheldon Axler. $\lambda$ is an eigenvalue of T if and only if $T-\lambda \mathrm{I}$ is not injective. I'm a bit confused about what ...
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0answers
16 views

How to understand the meaning of 'Oblivious' in Oblivious Subspace Embedding?

For the definitions of Oblivious Subspace Embedding and Subspace Embedding, please refer to the 1st page of paper http://arxiv.org/pdf/1308.3280v1.pdf.
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1answer
14 views

Vector notation for sum over elementwise product of 3 vectors

If I have an expression for two vectors $A$ and $B$ as below: $$\displaystyle \sum_{i=1}^N A_i B_i $$ we can write this as $ A^T B $ or $B^T A$ Now, if I have 3 vectors $A$, $B$ and $C$, ...
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1answer
34 views

Recover the inverse after interative solution of a linear system

I have solved the linear system $\mathbf{A} \mathbf{x} = \mathbf{b}$ with an iterative solver. The problem is well-posed ($\mathbf{A}$ is invertible, $\mathbf{b} \ne \mathbf{0}$, blah blah blah). ...
1
vote
1answer
30 views

Derivations on matrix algebra

Let $M=M_2(\mathbb{C})$ and let $\delta:M \mapsto M$ be a $\mathbb{C}-$linear map such that $\forall a,b \in M$ we have $\delta(ab)=\delta(a)b+a\delta(b)$. Prove that $\delta$ is of the form ...
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votes
0answers
27 views

Norm of a linear function [on hold]

$f : V \to V$ is a linear function. The basis of $V$ is $v_1, v_2, v_3$ Suppose $$\begin{align} f(v_1) &= 3v_2 \\ f(v_2) &= -5v_2 \\ f(v_3) &= 2v_1\end{align}$$ The norm of all basis ...
0
votes
0answers
29 views

Does isometry preserve volume on open sets?

Suppose there are two open sets $A,B$. $h$ is an isometry. And the function $h$ maps $A$ to $B$; $h(A)=B$. I need to show that isometry is volume preserving. Any hint would be appreciated! Thanks ...
2
votes
0answers
29 views

Rank of two bases

Suppose V and V' are finite spaces and A is the matrix of $\phi$ of whatever of two basis of V and V'.prove r($\phi$)=r(A).Now if we have basis $e_1 ... e_n $ the rank(A) is equal to the columns or ...
0
votes
1answer
55 views

Proving Submultiplicativity on a Matrix Norm

Let $||A||=(\sum_{i=1}^{n}\sum_{j=1}^{n}{a_{ij}^p})^{1/p}$, and let p=2. Then prove that $\|AB\|\le \|A\|\|B\|$ I have looked at numerous proofs for this, and I don't see one that satisfies me ...