Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Related to linear equation with one variable

My question is- Solve completely: $$ ax + b - \frac{5x + 2ab}{5} = \frac{1}{4}$$ Any guidance to solve this question would be helpful.
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3answers
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Inverse of transformation matrix

I am preparing for a computer 3D graphics test and have a sample question which I am unable to solve. The question is as follows: For the following 3D transfromation matrix M, find its inverse. Note ...
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Solving $11x-2=kx+15$ for a positive integral solution for $x$ given that $k$ is an integer

My question is- Find the integer value of $k$ such that the equation $11x-2=kx+15$ has positive integral solution for $x$. Find that solution. Any guidance to solve this question would be helpful.
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3answers
142 views

linear equation in one variable

My question is - Find the values of $a,b$ if the equation $a(2x+3) + 3bx=12x+5$ has infinitely many solutions. Please can anyone guide me to solve these types of questions? I would be really ...
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1answer
458 views

Decomposition of a symmetric matrix into multiplication of two vectors

What is the necessary condition for a real symmetric matrix $ A_{m\times m} $ to be written as $B*B^T$ where $B$ is an $(m\times 1)$ matrix ?
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325 views

Interpreting the determinant of matrices of dot products

In the Euclidean space $\mathbb{R}^n$ consider two (ordered) sets of vectors $a_1 \ldots a_k$ and $b_1 \ldots b_k$ with $k \le n$. Question What is the geometrical interpretation of ...
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120 views

Infinite dimension vector space decomposes into countable union of subspaces

The following problem is from Golan's linear algebra book. I have posted a solution in the answers. Problem: Let $V$ be a vector space over a field $F$ having infinite dimension over $F$. Show there ...
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140 views

Is the set $\{\frac{1}{a\,-\,\pi}\mid a\in\mathbb{Q}\}$ linearly independent over $\mathbb{Q}$?

The following problem is from Golan's linear algebra book. I have posted a proposed solution in the answers. Problem: Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Is the subset ...
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53 views

Witt Cancellation over $\mathbb{Z}/{p^e \mathbb{Z}}$?

I wonder whether someone knows if the Witt cancellation theorem also holds for the rings $\mathbb{Z}/{p^e \mathbb{Z}}$ where $p$ is an odd prime and $e \in \mathbb{N}$, i.e. for example, let $G = ...
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92 views

Dimension of gradient operator

Suppose \begin{align} y&=Ax\\ z&=f(y) \end{align} Then, is it true that: \begin{align} \nabla_xz&=\nabla_yf(y)\nabla_xy \\ &=\nabla_yf(y)A \end{align} Dimensions: $A: m \times n$ ...
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402 views

Counting invariant subspaces of a Vector space

Well, I was reading about invariant subspaces and related things and this question came to my mind: If I choose a vector space and fix a linear transformation on itself, then how many invariant ...
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Are the functions $\sin^n(x)$ linearly independent?

The following problem is from Golan's linear algebra book. I have posted a proposed solution in the answers. Problem: For $n\in \mathbb{N}$, consider the function $f_n(x)=\sin^n(x)$ as an element of ...
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2answers
887 views

How to prove that the union of two subspaces must be subsets of each other? [duplicate]

Possible Duplicate: Union of two vector subspaces not a subspace? $U,W\subseteq V$ are subspaces. Prove that in order for $U \cup W$ to be a subspace as well, either $U\subseteq W$ ...
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1answer
95 views

Is this function convex when the input vector is positive?

I am wondering if $f(\mathbf{x})$ is convex on the input of a vector of $n$ positive reals $\mathbf{x}$: $$f(\mathbf{x})=\operatorname{Tr}[(\mathbf{A}+\operatorname{diag}(\mathbf{x}))^{-1}]$$ where ...
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126 views

a minimum and maximum value problem

First and foremost, I greatly appreciated the prior attempts made by the excellent mathematicians Robert Israel, and mixedmath on the related problem. Now I have the following problems of the ...
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2answers
93 views

Introduction to Vectors

I am trying to write a hook for vectors on a linear algebra course. Does anyone have an opening hook for a section on vectors that will have a real impact on students?
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1answer
63 views

Proving $A\in M_n(\mathbb{R})$ is orthogonal

In an attempt to show that if $f(v)=Av$ is an isometry implies that $A$ is orthogonal I wish to show that $\forall x,y : \langle x-y,x-y \rangle=\langle A(x-y),A(x-y) \rangle \implies A$ is orthogonal ...
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111 views

A hard proof of two matrix's elements

This is not duplicate of A matrix's element proof, but it is harder than that one. Given an constant $\alpha \in (0,1)$, and an $n \times n$ matrix $X$ whose all entries are between 0 and ...
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A matrix's element proof

Thanks again for copper.hat and Robert Israel's quick immediate reply. While I am modifying the questions, they've already given the answer. Now in this thread, I've changed it back to the original ...
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1answer
139 views

Finding the eigenvalues of the sum of circulant and diagonal matrices - What am I doing wrong?

Saw this question about the eigenvalues of the sum of circulant and diagonal matrices on MO and, since I recall my prof mentioned circulant matrices and Robert Gray's book, I thought I'd give it a ...
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752 views

Prove $\mathbb{Z}$ is not a vector space over a field

This is an exercise from Chapter 3 of Golan's linear algebra book. Problem: Show $\mathbb{Z}$ is not a vector space over a field. Solution attempt: Suppose there is a such a field and proceed by ...
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1answer
311 views

Generalized Eigenvalue Problem

Consider a generalized Eigenvalue problem $Av = \lambda Bv$ where $A$ and $B$ are square matrices of the same dimension. It is known that $A$ is positive semidefinite, and that $B$ is diagonal with ...
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236 views

Orientation matrix for three points in the plane

In this Wikipedia entry for determining the orientation of a simple polygon, the following explanation is given for one of the steps: In computations, the sign of the smaller angle formed by a ...
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1answer
44 views

Matrix Algebra Help

Can someone please explain to me how for $Z$, $V \epsilon S^n$, $t \epsilon R$ $$(Z+tV) = Z^{1/2}(I + tZ^{-1/2}VZ^{-1/2})Z^{1/2}$$ I think I'm missing some fundamental aspect of matrix algebra. I've ...
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246 views

how to prove fact about linear combinations of basis vectors using relationship between matrix columns?

How do we prove that if $\{v_1,...,v_m\}$ and $\{w_1,...,w_m\}$ are bases for a real vector space, then there are at most $m$ real numbers $\lambda$ such that $v_1+\lambda w_1,...,v_m+\lambda w_m$ are ...
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141 views

Dimension of spaces of bi/linear maps

For $V$ a finite dimensional vector space over a field $\mathbb{K}$, I have encountered the claim that $$ \dim(\mathrm{Hom}(V,V)) = \dim(\mathrm{Hom}(V \times V, \mathbb{K})) $$ where ...
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1answer
96 views

MOD functions spanning the vector space $\mathbb{R}^{\{0,1\}^n}$

Let $MOD_{a,c}^r:\{0,1\}^n\to\{-1,1\}$ denote the function $$MOD_{a,c}^r(x)=\cases{-1 \ a\cdot x+c\equiv0\ (\text{mod r}) \\ 1 \ \text{else }}$$ Here $\cdot$ is the usual dot product. I want to ...
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44 views

Linear Algebra Order

I normally start with Gaussian Elimination, vectors and then matrices. I know a number of books which start with matrices and then go on to Gaussian Elimination and vectors. What is the most ...
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1answer
126 views

One-parameter subgroup of SL(n) is diagonalizable

Given a group homomorphism $\rho\colon \mathbb{C}^\times \rightarrow SL(n,\mathbb{C})$. Why is it true that the subgroup $\rho(\mathbb{C}^\times)$ is diagonalizable, i.e. that there is a basis of ...
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1answer
127 views

$Sp(A)\bigoplus Sp(B) \Leftrightarrow A\cup B$ is linearly independent

Is the following statement true? Assuming $V$ is a Vector Space and $A,B\subseteq V$ and that $A$ and $B$ are linearly independent: $Sp(A)\bigoplus Sp(B) \Leftrightarrow A\cup B$ is linearly ...
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1answer
176 views

Any permutation in symmetric group n can be rewritten as a composition of transpositions

I just want to show that a permutation can be written as a composition of transpositions. I cannot use cycles.
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Existence of a symplectic orthogonal transformation between A and B

Let's consider two symmetric real matrices $A$ and $B$ of dimension $2N$ and with the same (algebraic) eigenvalues, possibly degenerate. Is there a simple criterion to tell whether there exists or ...
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the asymptotic cone of matrices

I am just curious about something. The asymptotic cone of $\mathbb{Z}^n$ minus one point is $\mathbb{R}^n$. In fact, I think the asymptotic cone of $\mathbb{Z}^n$ minus a finite number of points is ...
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Give the parametric eqns of the line intersecting the planes

Give the parametric equations of the line of intersection of the planes $$4x + 2y + 2z = -1$$ and $$3x + 6y + 3z = 7$$ Also, give the equation of the plane that passes through the point $(2,-1,4)$ ...
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What's so special about the 4 fundamental subspaces?

I was reading Gilbert Strang's book for Linear Algebra (along with his lectures) and I feel that he is emphasizing that the 4 fundamental subspaces (Column Space, Row Space, Null Space and Null Space ...
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Find an equation for the plane that contains the following line and passes through point P

How do you determine the plane which contains the line \begin{align} x & = -1 + 3t \\ y & = 5 + 2t \\ z & = 2 + t \end{align} and passes through the point $P = (2,4,-1)$?
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Calculating Line Intersection with Hypersphere Surface in $\mathbb{R}^n$?

Given an (infinitely long) line described by two distinct points it contains in $\mathbb{R}^n$ $$\alpha^{(1)} = (\alpha_1^{(1)}, \alpha_2^{(1)},\dots,\alpha_n^{(1)})$$ $$\alpha^{(2)} = ...
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1answer
70 views

Proof that the linear map $T:\mathbb{R^n} \to\mathbb{R^m}$ is bounded

I would just like to know if my following proof is correct: Claim: If $T:\mathbb{R^n} \to\mathbb{R^m}$ is a linear map, then there exists $C > 0$ such that for every $x \in \mathbb{R^n}$$\|T(x)\| ...
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2answers
247 views

Finding the dimension of vector space V

I am trying to find the dimension of the vector space ($P_2$ means polynomials of degree at most $2$) $$V = \{p(x) \in P_2 \mid xp'(x) = p(x)\}.$$ However, I don't even know how to start... ...
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3answers
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Solving a simple 2 variable equation

I really forgot how to solve a equation like: $$-4A-3B=-2$$ $$3A-4B=0$$ I tried hard and I got $A=24/25$ and $B=24/75$. Can someone please verify this result? Also would be nice if you can show me ...
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1answer
457 views

Does null space always exist for a transformation?

This is inspired from this post as I was mentally playing with the concepts. The statement is the same just the transformation different, though for the benefit of everybody, I am repeating it, with a ...
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1answer
385 views

Vector space generated by the tensor products of pauli matrices

Let $\sigma_0,\sigma_x,\sigma_y,\sigma_z$ stand for the $2\times 2$ identity matrix and the well known pauli matrices: \begin{equation} ...
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50 views

A pdf on power of two having as mean a power of two

Given $k$, find $n \in N$ and $p_i$ such that $$\sum_{i=0}^n p_i 2^i = 2^k$$ $$\sum_{i=0}^n p_i = 1$$ $$0<p_i<1$$
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561 views

Null Space of Differentiation Transformation

What is the null space of differentiation transformation: $\frac{\mathrm{d} }{\mathrm{d} x}:P_{n} \to P_{n}$ where $P_{n}$ is the space of all polynomials of degree $\leqslant n $ over the real ...
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704 views

basis for hermitian matrices

let us consider $n\times n$ hermitian matrices. They form a real space. Now we know that any such matrix $A$ can be written as $A=A_+-A_-$, where $A_\pm$ are positive semidefinite matrices. Thus we ...
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103 views

Simple vector inequality

I was doing a long proof just now that had assumed that $||x|| - ||y|| \le ||x+y||$ I thought that I had done this before, but what I had done was actually unrelated: $||x-y|| \le ||x||+||y||$ ...
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2answers
283 views

A finite graph G is $d$-regular if, and only if, its adjacency matrix has the eigenvalue $λ = d$

Show that a graph $G$ finite with $n$ vertices is $d$-regular if, and only if, the vector with all the coordinates equals to 1 is eigenvetor from eigenvalue $λ = d$ from the adjacency matrix $A$ ...
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1answer
193 views

Inner Product and Gram-Schmidt Question

If $a \in \mathbb{R}$, let $\beta_a:\mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}$ be the billinear map defined by $$\beta_a \left( \left( \begin{array}{ccc} x_1 ...
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Finding the Dual Basis

Define the four vectors in $\mathbb{R}^4$ by $$v_1=\left( \begin{array}{ccc} 1 \\ 0 \\ 0 \\ 0 \end{array} \right), v_2=\left( \begin{array}{ccc} 1 \\ 1 \\ 0 \\ 0 \end{array} \right), v_3=\left( ...
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0answers
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Classifying A Matrix - matrix representation of an operator with linear operators as entries

Say that one has a matrix representation of an operator A with differential operators as entries in the matrix A. Is this a ...