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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Alternative proof of a transpose property

I am asked to prove; $$(AB)^T=B^TA^T$$ although it is very simple to prove it by the straight forward way, in the exercise I am asked to prove it without using subscripts and sums, directly from the ...
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1answer
45 views

What is the derivative of this? [duplicate]

I have a function of the following form: $J = \|W^TW-I\|_F^2$ Where, $W$ is a matrix and $F$ is the Frobenius Norm. How can I find the derivative of $\frac{\partial J}{\partial W}$ ?
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1answer
38 views

Computation rules for tensor products and inner products

I'm studying distributions and just came along the formula $$\langle f\otimes g,\phi\otimes\psi\rangle=\langle f,\phi\rangle\langle g,\psi\rangle.$$ I understand what that means in the context of ...
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3answers
202 views

I cannot make the mental leap from a vector to a function!

In my linear algebra book, it says that a vector is linearly independent if $\vec V = c1*\vec T_1 + c2*\vec T_2$ And then it goes on to say that $y(t) = c1 * e^{-at} + c2*e^{-bt}$ is linearly ...
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1answer
11 views

Find the matrix of a particularly defined endomorphism

Let $f$ endomorphism on $\mathbb{R^3}$ such that $-1$ is the only eigenvalue and $B={v,u,w}$ is a basis such that $f(v)=-v$ and $f(u)=-u$. I see that this should be an automorphism (otherwise it ...
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1answer
25 views

Closed vector space and a subspace of a vector space [duplicate]

What is a closed operation in a vector space? I don't see any difference between a closed operation in some vector Space R$^n$ and the open operation. What I mean by the closed operation is addition ...
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1answer
18 views

$W$ is a subspace of a given vectorspace $V$ [on hold]

$W$ is a subspace of given vector space $V$? $$V= R^2,\,\text{ and } W=\{(a,b):a,b\in\Bbb R, a\ge b\}$$
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1answer
25 views

Help to understand the basis for a dual space

I've been introduced to the concept of dual space in linear algebra. I can understand perfectly that the dual space of the space $V$ is a space $V^*$ made of all possible linear maps from $V$ to ...
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3answers
778 views

Any ideas on how I can prove this expression?

I don't have a lot of places to turn because i am still in high school. So please bear with me as i had to create some notation. In order to understand my notation you must observe this identity for ...
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0answers
17 views

How can I prove injectivity of this function

How can I prove that this function is injective: $f(x) = \dfrac{x(x+2014)}{\gcd(x, x+2014)}$ Domain and codomain: strictly positive natural numbers Where $\gcd$ is the greatest common divisor. I ...
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1answer
14 views

How can I find line segment connecting two vectors?

Let $S$ be a subset of $\mathbb{R}^n$. it is called convex if for all pairs of $a$, $b$, line segment from $b$ to $a$ is element of $S$. And it is given that $at+(1-t)$ is line segment between two ...
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1answer
38 views

What is the derivative of this?

I have a function of the following form: $J = \|W^TW-I\|_F^2$ Where, $W$ is a matrix and $F$ is the Frobenius Norm. How can I find the derivative of $\frac{\partial J}{\partial W}$ ?
2
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2answers
24 views

Definition of an image of a linear transformation

I have the following definition of an image of a linear transformation, but I think that there's a mistake in the size of the field (confusion between m and ...
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2answers
335 views

show projection matrix is equal to matrix times its transpose

Let $V$ be an $n$-dimensional real inner product space and let $a=\lbrace v_1,v_2,\dots v_n \rbrace$ be an orthonormal basis for $V$. Let $W$ be a subspace of $V$ with orthonormal basis $B = \lbrace ...
2
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0answers
29 views

Prove that $U_1\cup U_2$ is a subspace of $V$ $\iff$ $U_1\subseteq U_2$ or $U_2\subseteq U_1$ $\triangle$

Let $V$ be a vector space over some field. Let $U_1$ be a subspace of $V$. Let $U_2$ be a subspace of $V$. Prove that $U_1\cup U_2$ is a subspace of $V$ is equivalent to $U_1\subseteq U_2$ or ...
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1answer
101 views

Show that A and B commute

Let $A$ be a Hermitian matrix. Suppose there exists a matrix $B$ such that $A^3$ and $B$ commute. Show that A and B commute.
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0answers
15 views

Proving that $U \ \backslash \ \left\{u\right\}$ is complete

Let $V$ be a vector space over a field $F$. How to prove that if a system of vectors $U$ is complete and some vector $u ∈ U$ can be expressed as a linear combination of the vectors in $U \ \backslash ...
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1answer
17 views

Field extension of a vector space

If $V$ is a vector space over the field $k$, and $K$ is a field extension of $k$, then $(V)_K$ over $K$ is a vector space. How this new vector space is constructed? and how are the linear ...
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0answers
5 views

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
402 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
63 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
9 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 ...
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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 ...
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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
30 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 ...
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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 ...
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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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46 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 $2^ {n log (n)}$ ...
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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 = ...
5
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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 ...