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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Proof of a Vector Space

Let $F$ be a field and let $(V, +, F)$ be a vector space over $F$. If $W_1$ and $W_2$ are subspaces of $F$, prove that $W_1 - W_2 = \{v \in V | v = w_1 - w_2 \text{ for some } w_1 \in W_1, w_2 \in W_2 ...
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Understanding first part of dual basis proof

The textbook I'm reading attempts to proof the following: given $\left\{v_1, \ldots, v_n \right\}$ a basis for a vectorspace $V$ over $K$, there exists a basis $\left\{ \phi_1, \ldots, \phi_n ...
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transformation matrix between two different basis

I am working on this problem:- A rectangular coordinates $(x,y,z)$ are given in terms of new coordinates $(q_1,q_2,q_3)$ by :- $x=q_1 +q_2 \cos(\theta)$ , $y=q_2 \sin(\theta)$ and $z=q_3$. where ...
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If $A,B \in \mathcal{M}_n$ and $AB=BA=\mathbb{O}_n$ prove that $(A+B)^ν = Α^ν + Β^ν$.

I have one exercise in Linear Algebra and I would like to know if my solution is correct. If $A,B \in \mathcal{M}_n$ and $AB=BA=\mathbb{O}_n$ prove that $(A+B)^ν = Α^ν + Β^ν$. My first thought is ...
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Legendre transform is everywhere finite iff f grows faster than 2-norm

Let $f:\mathbb{R}^n \to \mathbb{R}\cup \{ \infty \}$ be convex. Its Legendre transform is $f^* (d):=\sup_{x\in \mathbb{R}^n}(d^Tx-f(x))$ Show $f^*(d)<\infty$ $\forall d\in \mathbb{R}^n$ iff ...
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Variable value estimation for given product/fracture values

I have a data set (time series) with given values for certain fractions xy = x/y (where x,y are not constant over time) Thus, there are following fractions: AB = A/B CB = C/B AD = A/D CD = C/D AE = ...
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Proof of Simple Properties of Volume

Let $e_{1},\ldots,e_{n}$ be vectors in $\mathbb{R}^{n}$. Define a parallelepiped $P$ to be a translate of the set $$\left\{x\in\mathbb{R}^{n} : x=t^{1}e_{1}+\cdots+t^{n}e_{n}, 0\leq t^{i}\leq ...
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A question on numerical range

Let $A,B \in {C^{n \times n}}$ and ${\sigma (A + B)}$ is spectrum of $(A+B)$. Suppose $M = \left\{ {\lambda \in C:\lambda \in \sigma (A + B),\left\| B \right\| \le \varepsilon } \right\}$ $F(A) = ...
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3answers
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Trouble with understanding dual space $V^{*}$.

I've just began the study of linear functionals and the dual base. And this book I'm reading says the dual space $V^{*}$ may be identified with the space of row vectors. This notion seems very ...
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How numerical radius help us to conclude an operator is normal and partial isometry?

In Furuta's book, "Invitation to Linear Operators" there is a theorem, theorem 2 in 3.7.3, that says: If $T^k=T$ for some integer $K\ge 2$ and if $w(T)\le 1$, then $T$ is the direct sum of a unitary ...
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968 views

Gauss-Seidel method convergence algorithm

From Wikipedia: The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either: ...
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2answers
31 views

Let $V=\bigcup_{i=1}^n W_i$ where $W_i$ s are subspaces of a vector space $V$. Show that $V=W_r$ for some $1 \leq r \leq n$.

Let $V=\bigcup_{i=1}^n W_i$ where $W_i$ s are subspaces of a vector space $V$ over an infinite field $F$. Show that $V=W_r$ for some $1 \leq r \leq n$. I know the result "Let $W_1 \cup W_2$ is a ...
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1answer
26 views

Linear algebra: generalize from characteristic $0$ a problem about polynomial coefficients.

Let $K$ be a field, and let $F$ be a subfield of $K$. Assume that $F$ is infinite. Let $p(x)$ be a polynomial in one variable with coefficients in $K$, and suppose that $p(a) \in F$ whenever $a \in ...
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Orthonomal bases and cross products

I want to show that if I have an orthonormal basis of $\mathbb{R}^3$, say $\{\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}\}$, and if $\boldsymbol{u} × \boldsymbol{v} = \boldsymbol{w}$, then we have ...
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1answer
16 views

Linear transformations for fixing the line $y = 0$

The professor says that the subgroup for "stabilizing" the line $y = 0$ is $$A = \begin{bmatrix} a & c \\ 0 & d \end{bmatrix}$$ because in order to fix the first basis vector, $b = ...
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41 views

generalized Cauchy-Schwarz inequality

How to prove $A'B(B'B)^{-1}B'A \leq A'A$, where $A$,$B$ are $n\times k$ matrices and $B'B$ is assumed to be positive definite? I don't see why it is a Cauchy-Schwarz inequality.
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If $L(v_1)=L(v_2)=L(v_3)=w_1$ then what is the $rank(L)$?

it is an elementary question. $L:V\to W$ is a linear transformation and $S=\{v_1,v_2,v_3\}$ is an ordered base of V, and $T=\{w_1,w_2\}$ an ordered base of W. If $L(v_1)=L(v_2)=L(v_3)=w_1$ what is ...
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30 views

Classification of all conjugacy classes of $GL_2(\mathbb{R})$, $GL_2(\mathbb{Q})$.

Give a classification of all conjugacy classes in the following groups. $GL_2(\mathbb{R})$ $GL_2(\mathbb{Q})$ My progress so far. If the characteristic polynomial splits, the matrix ...
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1answer
16 views

Approximate solution to a matrix equation

Let $A$ and $B$ be $n \times m$ matrices. I am looking for a $m \times m$ matrix $X$ which would be an approximate solution to the equation $AX = B$ (an exact solution is very unlikely to exist). More ...
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Matrices of Ordered Bases

Let $V$ be a real finite-dimensional vector space and $T : V → V$ be a linear map. Let $E$ be a basis of V . What does it mean to say that $A$ is the matrix of $T$ with respect to $E$. Let $S : V → V$ ...
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29 views

about the complex conjugation of a complex vector space $V\left(\mathfrak{\mathcal{C}}\right)$

I am a little confused about the complex conjugation of a complex vector space $V\left(\mathfrak{\mathcal{C}}\right)$. From other answers (Is a complex vector space closed under complex ...
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How are closed form solutions for eigenvalues constructed from sines and cosines?

For example a size N 3-point finite difference scheme has eigenvalues $\lambda_j = 2+cos(j\pi/(N+1))$. How is this determined? I know the Gershgorin circle theorem, but this is not what I am looking ...
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2answers
338 views

Multiplying Out Inner Products

If I have a product of the form $(x-s)^tA(x-s)$ where $x$ and $s$ are vectors and $A$ is a matrix, how would I go about multiplying this out? Further, how would I go about taking its derivative?
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$(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse?

Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) ...
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1answer
30 views

Do approximate identities remain approximate identities if one adjoins 1 to a C* Algebra?

If we have a C* Algebra $\mathscr{U}$ without an identity we can adjoin an identity $\mathbb{1}$ in the following way: We take $\mathscr{\tilde U}$ to be the set $\{(\alpha,A); \alpha \in ...
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632 views

Eigenvalues and eigenvectors of Hadamard product of two positive definite matrices

The component-wise product (Hadamard product) of two positive definite matrices is a positive definite matrix (Schur product theorem). I encountered the following proof of it: $A=(a_{ij})$ and ...
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2answers
56 views

Number of matrices $A \in M_n(\mathbb{F}_q)$ where $A^2 = 0$.

What is the number of matrices $A \in M_n(\mathbb{F}_q)$ for which $A^2 = 0$ (as a function of $n$ and $q$)?
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2answers
30 views

Is $V = \{(x,y,z)\in \mathbb{R}^3:\ x+y >1 \}$ a subspace?

Prove whether the following subsets of $\mathbb{R}^3$ are subspaces : (a) $$V = \{(x,y,z)\ \in \mathbb{R}^3:\ x+y >1 \ \},$$ I think that this is not a subspace as the zero vector does not ...
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Identification between wedge product and its dual

Let $\mathbb{F}$ be a field, and let $(e_i)$ be the usual elementary basis of $\mathbb{F}^n$. Let $\varphi_{ij}: \mathbb{F}^n \wedge \mathbb{F}^n \to \mathbb{F}$ be such that $v \wedge w \mapsto ...
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1answer
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Relation between Positive definite matrix and strictly convex function

I have a problem. From wikipedia http://en.wikipedia.org/wiki/Positive-definite_matrix any function can be written as $$z^TMz$$ where z is a column vector and M is a symmetric real matrix. However ...
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2answers
47 views

Can such an “orthogonal” matrix exist?

I know that the definition of an orthogonal matrix is that $A \in \mathbb R^{n \times n}$ is orthogonal if $AA^T = A^T A=I$, no problem with that whatsoever. My question is this - Why only square ...
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Linear or Nonlinear function?

I want to know if the below two function are Linear or not. I have been searching in Google but found some confusing results. The first one is an inequality and the second one is just a function and ...
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What are the (more general) conditions for consistency in a system of linear equations?

Usually, when speaking about conditions for the existence of solutions to linear equations of the form $A x = b$ (with $A \in \mathbb{R}^{n \times n}$ and $x , b \in \mathbb{R}^n$), one is interested ...
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1answer
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Finding Marginal Density functions with $Y\sim N_4(\mu,\Sigma)$

Suppose $Y$ is $N_4(\mu, \Sigma)$ where $$\mu = ( 1,2,3,-2)'$$ and $$\Sigma =\begin{bmatrix} 4& 2& -1& 2 \\ 2& 6& 3& -2 \\ -1& 3& 5& -4 \\ 2& ...
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1answer
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Given $k$ distinct linear operators, prove such an $\alpha$ exists

I have $k$ distinct linear operators $\{\phi_i\}$ which act on $V$, a vector space on some number field $K$ (in the sense that $\Bbb Q$ is the smallest possible one). Now I have to prove that there ...
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1answer
46 views

How to solve these equations?

How to solve these equations for a, b, c and x? I have the following: $ 2a+b+c = 1$ $a = (a+b)x + 0.25(a+c) $ $a=(a+c)(1-x)$ $b=a(1-x)+c(x-0.25)$ $c=b(1-x)+a(x-0.25)$ I tried, but ended ...
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tangent plane for y^x at point (2,1)

I test my answer using wolfram alpha pro but it gets a different result to what I am getting. This is homework. My result is z= 2(y-1) partial derivative with respect to y is ...
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1answer
18 views

How to find the minimum sum of unknown variables that is a solution to a system of two linear equations?

I'm trying find the minimum sum of $x_{1} + x_{2} + ... + x_{n}$ where these are a solution to a linear system of two equations. System of linear equations in general form: $$ a_{11}x_{1} + ...
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algebra questions [on hold]

On the weekend you played Rugby League. You scored three tries, made two conversions and one field goal. How many points did you gain for your team in total? Write a general equation for this problem. ...
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Seemingly impossible problem involving linear combination of vector components.

Express $\langle 4, -8 \rangle$ as a linear combination of $\vec{u}$ and $\vec{v}$, given $\vec{u}=\langle 1,1 \rangle$ and $\vec{v}=\langle -1,1 \rangle$. So, I set up: $\vec{i}=\langle 1,0 \rangle$ ...
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1answer
14 views

Project point on plane - Unique identfier?

I have a number of planes (in $\mathbb{R}^3$), each represented by a point $\vec{P_i}$ which lies within each plane and the normal vector $\vec{n_i}$. If I project a point $\vec{Q}$ (which does not ...
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2answers
37 views

Is $V$ a simple $\text{End}_kV$-module?

Let $V$ be a finite-dimensional vector space over $k$ and $A = \text{End}_k V$. Is $V$ a simple $A$-module?
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Quadratic form in Hilbert space associated with orthogonal projection operator

we are in Hilbert space $L^2 $ and we are given subspace of dimension $2K$ $$ V=Vect\{ g_k,\bar{g_k},1\le k\le K \}$$ $V$ is a sum of $K$ subspaces of dimension 2 $$ W_k=Vect \{g_k,\bar{g_k} \} $$ now ...
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Square root of differentiation

Let $T=d/dx$ be the differentiation operator on vector space $V=C^{\infty}(\mathbb{R})$, the space of real (complex) valued smooth maps on real line. To what extent, all subvector space ...
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2answers
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What am I doing wrong in finding the orthogonal projection of a vector onto the subspace V?

Let $V∈ℝ^5$ be the subspace $V=span{(2,0,0,0,1),(0,2,0,3,0)}$ and let $w=(0,0,-4,-1,-1)$. Find the orthogonal projection of $w$ onto V,using exact values in your answer. My Approach Let the ...
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1answer
10 views

Isolate Costs in NPV equation

Hey can anyone help with this? This is the classic NPV equation: NPV = -CapEx + ∑ (Revenue − Costs) / (1+Discount)^i The partial sum is from i = 0 to n years. For my purposes all the elements ...
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1answer
27 views

Prove $v,w\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and dependent when $p=3$

I need to prove that $\{v=(6,9),w=(7,8)\}\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and linearly dependent when $p=3$. The problem is my freshman algebra course did not cover rings and ...
2
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0answers
14 views

dimension of intersection of subspaces

In a 13-dimensional vector space, the dimension of intersection of two 6-dimensional subspaces is at least 1 at most 1 at least 6 at most 6 My thought:For two subspaces U and W of vector space V, ...
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12 views

About finding the common diagonalizing similarity transformation.

Say I have $2k$ matrices $M_{a_1b_1}$, $M_{a_2b_2}$,..,$M_{a_kb_k}$ and their negatives. Here $M_{a_ib_i}$ is such that it has $0$ everywhere except that it has $1$ at $(a_i,b_i)$ and $(b_i,a_i)$ ...
2
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2answers
37 views

Prove function space is linearly independent.

Let $V$ the space of all funcions $f:Ŗ\rightarrow R$. Prove that the ten functions defined by $x\rightarrow |x-1|$,$x\rightarrow |x-2|$,....,$x\rightarrow |x-10|$ are linearly independent. I need ...