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 ...

learn more… | top users | synonyms

0
votes
1answer
22 views

array of $n$ numbers, find the $2$ missing numbers

Given an array of size $n$. It contains numbers $1$ to $n$. Each number is present at least once, except for $2$ numbers. What algorithm will allow you to find the $2$ missing numbers?
2
votes
0answers
8 views

extended PCA (tangled matrices)

Given an m by n matrix A and the constant r, the principal component analysis allows us to find matrices W and H so that the WH gives a lower rank approximation of A. In other words, $A_{m\times n} ...
1
vote
1answer
23 views

Finding linear independence in $v_1,\ldots,v_m$

First, I'll try not to ramble, although it tends to happen when I type. I have the following linear algebra problem for my homework. Prove or give a counterexample: If $v_1, v_2, \ldots , v_m$ are ...
3
votes
1answer
32 views

Real matrix with the property that every nonzero vector in $\mathbb{R}^n$ is an eigenvector of $A$.

so I'm supposed to let $A$ be a square real matrix with the property that every nonzero vector in $\mathbb{R}^n$ is an eigenvector of $A$. And I'm supposed to show that $A=\lambda I$ for a constant ...
-1
votes
0answers
3 views

How to variably average spreadsheet columns together in a ranking of priority?

I have a variable number of spreadsheet columns, ordered from most to least significant. I want to average them into a variable number of columns, giving priority to the leftmost significant columns ...
1
vote
1answer
12 views

Determining optimal size of rectangle to maximise volume

I am having problems understanding how to solve this question, any help would be much appreciated. An open box (no top) is made from a rectangular sheet of cardboard that measures 20 cm by 30 cm by ...
0
votes
2answers
32 views

Determine the kernel of a linear map $f:U \to V$

Let $U=<\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & -1 \\ 0 & 1 & 0\end{pmatrix}, \begin{pmatrix}2 & 0 & 1 \\ 0 & 1 & ...
0
votes
0answers
11 views

Can anyone help me prove derivative of scalar field using mean value theorem?

Assume that f′(x;y)=0 for every x in some n-ball B(a) and for every vector y. Use the mean value theorem to prove that f is constant on B(a). And if f′(x;y)=0 for a fixed vector y and for every x ...
1
vote
0answers
11 views

General form of an element of the othogonal basis of $q$

Let $$q \begin{pmatrix}a & b \\ c & d\end{pmatrix}= (a-b)^2+(b-c)^2+(c-d)^2$$ quadratic form on $M_2(\mathbb{R})$. How can I prove that every orthogonal basis $B$ of $M_2(\mathbb{R})$ has ...
0
votes
1answer
15 views

Diagonalizability of endomorphism $f:M_2(\mathbb{R}) \to M_2(\mathbb{R})$ such that $f(A)=tr(A)I_2-A$.

Let $f:M_2(\mathbb{R}) \to M_2(\mathbb{R})$ such that $f(A)=tr(A)I_2-A$. How can I determine what is the explicit expression of $f$, and, most importantly, how do I see if it is diagonalizable? The ...
1
vote
2answers
27 views

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 ...
1
vote
1answer
20 views

Clarifications of problem with parameters: the relationship between matrices and endomorphism

Let $f$ be an endomorphism of $R^3$ such that $f(a,b,c)=(2b,a-b,b)$. I don't understand how I can see for which values of $k\in R$ there esist $$\begin{pmatrix}-2 & 0 & 0 \\ 0 & k & 0 ...
1
vote
1answer
8 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 ...
2
votes
1answer
25 views

Trying to understand proof that 3 non-collinear points determine a unique plane

$Q,R,P$ are 3 non-collinear points. Plane $M = P + s(Q-P) + t(R-P)$. Let $C = Q-P$ and $D= R-P$. Let us grant that C and D are linearly independent. Let $M' = P + sA + tB$. Assume $M'$ has $P,Q,R$. ...
1
vote
1answer
23 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 ...
-1
votes
1answer
16 views

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

$W$ is a subspace of given vector space $V$? $$V= R^2,\,\text{ and } W=\{(a,b):a,b\in\Bbb R, a\ge b\}$$
2
votes
1answer
26 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 ...
-1
votes
0answers
16 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 ...
1
vote
1answer
21 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 ...
1
vote
1answer
13 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 ...
1
vote
1answer
32 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 $\frac{\partial J}{\partial W}$ ?
0
votes
0answers
8 views

Matrix function to express pair-wise distances of rows in $X, Y$

There are two real matrices: $X, Y$ with $X$ being of dimension $n_1$ x $p$, $Y$ of dimension $n_2$ x $p$. The goal is to form the matrix $D$ of dimension $n_1$ x $n_2$ where each element $d_{ij}$ ...
0
votes
0answers
13 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 ...
0
votes
1answer
43 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 $\frac{\partial J}{\partial W}$ ?
0
votes
0answers
4 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 ...
1
vote
1answer
15 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 ...
2
votes
0answers
27 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 ...
0
votes
1answer
22 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: $$u\in V,\ 0\cdot u=\vec{0}$$
0
votes
1answer
21 views

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

Let $F$ be a field. How to prove: $$\exists\alpha\in F,\ \alpha\cdot\beta=\beta,\ \beta\in F \Rightarrow\alpha=1$$
1
vote
1answer
38 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 $x=0$ , ...
2
votes
1answer
16 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 ...
0
votes
0answers
7 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 ...
1
vote
1answer
18 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 ...
0
votes
0answers
7 views

Issues with a particular bilinear form and determining rank, signature, etc. of its restriction

Let $b: M_2(\mathbb{R}) \times M_2(\mathbb{R}) \to \mathbb{R}$ such that $b(X,Y)=trace(X^tAY)$, where $X^t$ is the transpose of $X$ and $A=\begin{pmatrix} 2 & 1\\1 & 0\end{pmatrix}$. In my ...
2
votes
1answer
23 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 ...
0
votes
0answers
8 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 $m$ ...
1
vote
1answer
37 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 ...
0
votes
0answers
9 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 $K_p$ ...
1
vote
1answer
49 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?
0
votes
1answer
17 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 ...
4
votes
1answer
46 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 ...
0
votes
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 ...
1
vote
1answer
15 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 \\ ...
1
vote
1answer
28 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 ...
0
votes
0answers
33 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
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 ...
1
vote
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 ...
2
votes
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 = ...
1
vote
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
vote
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 $Mx_3$? ...