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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Minimal polynomial of a diagonal matrix

How can I show that the minimal polynomial of a diagonal matrix is the product of the distinct linear factors $(A-\lambda_{j}I)$? In particular, if we have a repeated eigenvalue, why is it that we ...
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29 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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24 views

The highest direction of the trace operator

Let $W$ be a real and symmetric matrix ${m \times m}$ from the set $\widetilde{W_m}$, and $T:\widetilde{W_m} \rightarrow \mathbb{R}$ a function defined by $T(W) = trace(W^3)$. We are interested to ...
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Proof that a set of vectors can be extented to a basis (but differennt from basis expansion theorem)

I have a set $S={v_i}$ of N vectors in a d ($d<N$) dimensional space (call it V), I know that those vectors span the whole space, but of course they cannot be a basis. I can expand the space to a ...
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2answers
45 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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33 views

Linear independent sets

Let $S_1\in\mathbb{R}^{n}$ and $S_2\in\mathbb{R}^{n}$ be two subspaces of $\mathbb{R}^{n}$ Suppose $x_1\in S_1$, $x_1\notin S_1\cap S_2$. $x_2\in S_2$, $x_2\notin S_1\cap S_2$. Show that $x_1$ and ...
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68 views

How to show that this symmetric matrix is invertible?

This problem is linked to a physical problem of linear system (structural analysis of a beam with $n+2$ supports), so this matrix $A$ should be invertible. The matrix is $n\times n$ and has elements ...
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1answer
13 views

Product of countably many 1-dimensional spaces does not have cardinality $\aleph_0$

From Bergman's "Universal Algebra: Fundamentals and Selected Topics" page 52, constructing a directly indecomposable algebra (one which does not admit a decomposition into directly indecomposable ...
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If $A: M \to M$ then $M$ is $A$-invariant subspace and $A $ is an endomorphism?

Just straightening out the terminologies here... Given If $A: M \to M$ then $M$, $M$ some subspace of a vector space, is the following statement equivalent: $M$ is a $A$-invariant subspace $A $is ...
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52 views
+50

Closed formula for Poincaré series in terms of adjacency matrix.

Let $Q$ be a finite quiver with vertex set $I$. For each $n = 0, 1, 2, \dots,$ let $k^{(n)}Q \subset kQ$ be the $k$-linear span of all paths of length $n$, in particular, we have$$k^{(0)}Q = ...
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30 views

Do addition and multiplication define a structure of a field? [duplicate]

I am taking an advanced linear algebra course for my Masters but never took linear in undergrad so please realize I know little to nothing about these topics. Question: Let r exist in R and 0 not ...
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5answers
36 views

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
771 views

Find the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$.

Find a vector equation and parametric equation of the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$. Find two points on the line that are ...
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2answers
33 views

How to solve for the matrix $X$ in the following equation $AXB + X = CD$

How to solve for the matrix $X$ in the following equation $AXB + X = CD$? $A$ and $B$ are full rank symmetric matrices, and there is no structure to $CD$. $CD$ just could be $C$.
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1answer
18 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 $(b_i,a_i)$ ...
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1answer
49 views

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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2answers
32 views

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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1answer
15 views

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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1answer
25 views

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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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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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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2answers
973 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: ...
2
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1answer
32 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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2answers
49 views

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
19 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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2answers
49 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 ...
5
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1answer
33 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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0answers
15 views

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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33 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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0answers
18 views

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
340 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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1answer
634 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 ...
2
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2answers
34 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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2answers
46 views

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
2k views

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
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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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76 views

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
28 views

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
14 views

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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2answers
20 views

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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0answers
19 views

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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1answer
19 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
39 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?
5
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0answers
53 views

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
15 views

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 ...
1
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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 ...