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

3
votes
0answers
19 views

Is this the correct solution involving vector subspaces and basis?

I need to find the basis and hence dimension of a subspace of $\mathbb{R^3}$. 1) $$U=\{(x,y,z):x=2y\}$$ Solution: We have $x=2y \iff y=\frac{x}{2}$ therefore we can write all elements in $U$ as the ...
3
votes
0answers
69 views

Matrix which is not similar to it's transposed

Let $V$ be vector space over a field $\mathbb{k}$. I can prove that any matrix is similar to its matrix transpose if $\mathbb{k}$ is an infinite field, but is this still true when $\Bbb k$ is finite? ...
3
votes
0answers
42 views

Matrix equation $A=PBP^{-1}$

Suppose $P$ is invertible and $A=PBP^{-1}$ . Solve for $B$ in terms of $A$. My attempt: I just left multiplied the equation by $P^{-1}$ and right multiplied it by $P$ so that I got $B=P^{-1}AP$. Is ...
3
votes
0answers
52 views

Which of the following subsets of $\mathbb C$ is a field?

I'm not entirely sure that I understand the concept of fields fully so I'll give you the question and then I'll let you pick my brain and tell me if my logic is correct. Please note: I'm not just ...
3
votes
0answers
43 views

problem book on linear algebra

please refer a problem book on 1.Linear algebra :TOPICS: Vector spaces, Linear dependence of vectors, basis, dimension, linear transformations,matrix representation with respect to an ordered basis, ...
3
votes
0answers
80 views

Convergence of integers by transformations

Let $x=(a,b)$, where $a,b$ are in $N$ Now we have the transformations: $$T_1(x) = (ka, b+1)$$ $$T_2(x) = (b,a)$$ where $k$ is in $N$. Where the order of choosing a transformation is not fixed. ...
3
votes
0answers
47 views

When does $\| \Pi \|_1 = 1$ where $\Pi$ is a projection.

By projection I mean any matrix such that $\Pi = \Pi^2$. It is well known that all projections can be written as $\Pi = A(B^\top A)^{-1}B^\top$ for some $A,B$. Characterize the class of projections ...
3
votes
0answers
45 views

Eigenvalues of Overlapping block diagonal matrices

I look for eigenvalues of general overlapping block diagonal matrices. e.g. $$\left[ \begin{matrix} 1 & 4 & 0 & 0 & 0 & 0\\ 4 & 2 & 3 & 2 & 0 & 0\\ 0 & 3 ...
3
votes
0answers
38 views

(Fast) eigen decomposition of $DXD$ where $D$ is diagonal, $X$ is symmetric with known eigen decomposition

Assuming that I already know the eigen-decomposition of a real symmetric matrix $X$, is there any way to use it to retrieve efficiently the eigen-decomposition of $DXD$, where $D$ is a diagonal ...
3
votes
0answers
43 views

The relation between $\inf_{R\in \mathsf{U}_n} \left\Vert A - BR\right\Vert^2_F$ and $\left\Vert AA^*-BB^*\right\Vert$

Suppose that $A$ and $B$ are two arbitrary $m\times n$ matrices with $m>n$. Let $\mathsf{U}_n$ denote the set of $n\times n$ unitary matrices. I'd like find positive constants $c_1$ and $c_2$ such ...
3
votes
0answers
77 views

quadratic form from nxn matrices to reals ( Tr(A^2) ). I need to find it's signature and rank.

Firstly prove $Tr(A^2)$ defines a quadratic form from the space of $n \times n$ matrices to R. I think you just have to show that $Tr(A B)$ is a bilinear form which seems too easy to be correct or I'm ...
3
votes
0answers
58 views

Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
3
votes
0answers
29 views

Multivariate polynomials with prescribed zeros

Let $P_k\subsetneq\Bbb R[x_1,\dots,x_n]$ be the set of degree $k$ multivariate real polynomials. Pick a subset $S$ of $\{-1,+1\}^n$ of size $|S|<\sum_{i=0}^k\binom{n}{i}$. We seek a polynomial ...
3
votes
0answers
41 views

Interpretation of a 3 Variable System of Equations

I'm a high school student, and, of course, this week is finals week. For my Algebra 2 semester final, we have been permitted to take the test home and collaborate with others. This final can be viewed ...
3
votes
0answers
49 views

Finding $n$ scalars such that $\det{(cI-A)}=0$ without eigenvalues

My problem is this Let $A$ be an $n\times n$ matrix over $\mathbb{F}$. Prove there are at most $n$ distinct scalars $c\in\mathbb{F}$ such that $\det{(cI-A)}=0.$ I know that the determinant is ...
3
votes
0answers
38 views

How do $\Bbb{R}$ and $\Bbb{R}^2$ have the same dimension over $\Bbb{Q}$ as vector spaces?

How do $\Bbb{R}$ and $\Bbb{R}^2$ have the same dimension over $\Bbb{Q}$ as vector spaces? Having the same dimension would imply that both have the same number of basis elements. How do I prove that ...
3
votes
0answers
50 views

Simultaneously diagonalizable without distinct eigenvalues

It is a well known result that if $u$ and $v$ are two diagonalizable endomorphisms of a $\mathbb{C}$ finite-dimensional linear space $E$, if $u$ (or $v$) has distinct eigenvalues and if $u$ and $v$ ...
3
votes
0answers
130 views

If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$

If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$ where $S$ is a invertible matrix and $B$ has the form $B = \left( \begin{array}{ccc} 0 & a_1 & 0 & ...
3
votes
0answers
74 views

Shortlist of problems in linear algebra

A while ago I remember seeing a very nice shortlist of problems in linear algebra. It was a list of about 40-50 problems. The idea was that if you solve them, you learn linear algebra very well and ...
3
votes
0answers
56 views

Can we show it without involving that $V=V^{**}$ are canonically isomorph?

My text proves the following Theorem. Let $V$ be a vector space over $F$ and $B=\{ v_1, \ldots , v_n \}$ a basis of $V$. Then there is exactly one basis $B^*=\{ f_1, \ldots , f_n \}$ of $V^*$ with ...
3
votes
0answers
98 views

Finding the new basis for a diagonalized quadratic form

I have been given the quadratic form $$A(x,x) = 2x^2-\frac{1}{2}y^2-2xy-4xz$$ and been asked to diagonalize it, find the change of basis matrix, and find the new basis in which A is diagonalized. I ...
3
votes
0answers
93 views

How to derive the following from Azuma's inequality?

This is claimed in Proposition 1 in the paper http://arxiv.org/abs/1409.6110 Let $A$ be a $n \times d$ matrix. $A$ can have only $K$ different types of rows i.e. rows of $A$ are chosen from a set of ...
3
votes
0answers
37 views

the exact graph of the general solution for $x'=\begin{bmatrix} 1 & 1\\ 4& 1 \end{bmatrix}x$

i need someone to give me exact graph of the general solution for $$x'=\begin{bmatrix} 1 & 1\\ 4& 1 \end{bmatrix}x$$ i solved it manually , the general solution is like this ...
3
votes
0answers
56 views

Proving that table totals can always be preserved with ceiling and floor

$\begin{array}{|c|c|c|c|} \hline 11.998& 9.083 & 2.919 & &24 \\ \hline 12.983&10.872&3.145&&27\\ \hline 1.019&2.045&0.936&&4\\ \hline & & ...
3
votes
0answers
104 views

Jordan normal form theorem proof question

Theorem: Assume that the characteristic polynomial $x_f$ splits into linear factors. Then there exists a Jordan normal form for f. The Jordan normal form is unique up to the order of the Jordan ...
3
votes
0answers
55 views

Find the projection of any vector onto the linear span and the normal from any vector to that span

Show that the vectors $u_1 = (1/9,4/9,8/9), u_2=(8/9,-4/9,1/9), u_3=(-4/9,-7/9,4/9)$ form an orthonormal basis of $\mathbb{R}^3$. Find the projection of any vector $x=(\xi_1,\xi_2,\xi_3) \in ...
3
votes
0answers
44 views

Adding Subpaces

Assume $F$ is a field, $V$ is a vector space over F, and $U$ is a subspace of $V$ . Assume $a \in F$. Prove that $U + (aU) = U$. Here is my approach: 1). Check $U+aU \subseteq U$ ?Suppose $u\in ...
3
votes
0answers
86 views

Random multipliers of polynomial values at known points in $\mathbb{Z}_p$

Scenario (1) We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$, where $p$ is a prime number. Let $P_i$ be a polynomial such that $P_i \in R[x]$. We ...
3
votes
0answers
56 views

subspace of the Vector Space of real valued functions

This is a problem from Hoffman and Kunze's Linear Algebra 2nd edition. I am trying to determine whether or not a particular subset of the set of all real valued functions is a subspace. I've done ...
3
votes
0answers
34 views

find these linear functionals

I'm trying to solve this question: My attempt of solution: If $x\in E$, see $x$ in the first $m$ coordinates of $\mathbb R^n$ (can we do this?). I know how to find linear functionals such that ...
3
votes
0answers
35 views

Linear code from larger linear code

Question 2.16 of Essential Coding Theory by Guruswami, Rudra and Sudan asks to produce a $[n - d, k - 1, d'\geq\lceil d/q\rceil]_q$ code from an arbitrary $[n, k, d]_q$ code. Here we are working over ...
3
votes
0answers
104 views

Tensor product of algebras which is Frobenius.

Let $A$ and $B$ be two finite dimensional algebras over a field $k$. Let us suppose that the $k$-algebra $A\otimes_{k} B$ is Frobenius (or symmetric). Is it true that $A$ and $B$ are two Frobenius ...
3
votes
0answers
62 views

Prove that the determinant of a given matrix is proportional to the area of the triangle whose corners are the three points.

For three points in 2D, $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$, show that the determinant of \begin{bmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 2\\ x_3 & y_3 & 3\\ ...
3
votes
0answers
50 views

Proof check: any linear transformation can be represented as a matrix-vector product

I'm trying to prove that Theorem. Consider a linear transformation $T : \mathbb R^n \to \mathbb R^n$. The transformation $T$ can be represented as a matrix product $\mathbf x \mapsto A \mathbf ...
3
votes
0answers
120 views

Algorithms for solving overdetermined, homogeneous linear systems with multivariate polynomial coefficients

I would like to solve overdetermined, homogeneous linear systems of equations with multivariate polynomial coefficients, i.e., $Ap=0$ with $A$ an $m\times n$ matrix, $m\gg n$, and $a_{i,j} \in ...
3
votes
0answers
173 views

Jacobian for a matrix transformation: Example of Cholesky decomposition

I would like to generally understand how the Jacobian of a matrix transformation can be computed. As a concrete example, consider the Transformation from a (correlation) matrix to its Cholesky ...
3
votes
0answers
84 views

Matrix partwise multiplication

I am working on an artificial intelligence application that (among other things) combines "opinions" of several "experts" who each have access to different aspects of a "situation". I can build this ...
3
votes
0answers
94 views

“Rank-K Correction” of a matrix and significance?

Today my studies led me to read about the matrix inversion lemma, which Wikipedia introduces as follows: In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max ...
3
votes
0answers
405 views

Vector spaces - Multiplying by zero scalar yields zero vector

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space related axioms. ...
3
votes
0answers
133 views

Proof of rank-nullity via the first isomorphism theorem

I was thinking about the proof of the rank-nullity theorem and I thought about proving it as follows. I just wondered whether this proof worked? Lemma. If $V$ is a finite-dimensional $F$-vector space ...
3
votes
0answers
47 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
3
votes
0answers
39 views

Does there exist a pair $(i,j)$ such that $x_{i}(1-x_{j})$ and $x_{j}(1-x_{i})$ are not greater than $\frac{1}{4\cos^{2}\frac{\pi}{n+1}}$?

Let $x_{1},x_{2},\ldots,x_{n}\in[0,1]$. Do there always exist $i,j$, $1\le i<j\le n$, such that both $x_{i}(1-x_{j})$ and $x_{j}(1-x_{i})$ do not exceed $\frac{1}{4\cos^{2}\frac{\pi}{n+1}}$?
3
votes
0answers
49 views

Bayesian linear regression cost function

I am studying classification using linear regression . Now, I want to map it in Bayesian regression. Let talk about binary classification using linear regression again. Assume that I have a set ...
3
votes
0answers
31 views

Finitely Generated Subset

Let $L\subset\mathbb{Z}^n$ be a subset closed under addition and subtraction. Can we show that $L$ is generated by at most $n$ elements?
3
votes
0answers
59 views

Show that a certain operator is symmetric

I am trying to prove that the operator $L^2 = -\partial_\theta^2 - \cot\theta\,\partial_\theta - \frac{1}{\sin^2\theta}\partial_\phi^2$ fulfills the following property: For $y_{l,m} = ...
3
votes
0answers
67 views

Integer Solutions To Linear Equation

$$a*q_1+b*q_2=c$$ $$a*q_3+b*q_4=f$$ $q_1, q_2, q_3, q_4$ rational numbers, $c,f$ integer Given $q_1, q_2$ can you construct all solutions $(a,b)$ where $c,f$ is intenger I made an edit since the ...
3
votes
0answers
143 views

Find minimal set of cosets $C$, so that each $2$ vectors in $A_n$ are in one coset in $C$

Let $F_2^n$ be the set of all vectors of length $n$ with values of $0$ or $1$ and $A_n$ = $F_2^n \setminus(11\ldots1)$. Set $A_n$ contains all vectors except one with all $1$s. We can consider cosets ...
3
votes
0answers
211 views

How prove this stronger Cauchy-Schwarz inequality for traces of compression matrices

Question: Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that: ...
3
votes
0answers
58 views

Does anyone know any reference for this matrix?

For $n \geq 4$, $A$ is $(n-1) \times (n-1)$ tridiagonal block matrix $$A = n^2 \begin{bmatrix}B & -I & 0 & \cdots & \\-I & B & -I & 0 & \\ 0 & -I & B & -I ...
3
votes
0answers
47 views

Solve the matrix equation $[I+X(I-ST)^{-1}S](I-TS) = I$

Suppose that $I, S, T$ and $X$ are linear transformations of the vector space $W$, that $I$ is the identity and that $I-ST$ is invertible. Solve the equation $$[I+X(I-ST)^{-1}S](I-TS) = I$$ for $X$ ...