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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2
votes
1answer
15 views

Show that $\{w_1,\dots,w_p,v_1,\dots,v_q\}$ is an orthogonal set and spans $\mathbb{R}^n$

These series of questions build up on each other i'm stucked on the last one, i'm also not sure if all of these work but I am pretty convinced they do. Let $W$ be a subspace of $\mathbb{R}^n$ with an ...
4
votes
0answers
28 views

Cauchy-Schwarz Inequality without using $\langle a x,y\rangle=a\langle x,y\rangle$

Can we prove the Cauchy-Schwarz Inequality for an inner product space without using $$\langle a x,y\rangle=a\langle x,y\rangle\,\,\,\,\ ?$$ If the answer is "NO" can we prove it?
0
votes
3answers
28 views

Dimensions of a basis of a coordinate space

I need a little clarification on the relationship between the basis, its dimension and their corresponding real coordinate space. Suppose we are operating in the fourth coordinate space ...
0
votes
2answers
25 views

What is the difference between orthogonal and orthonormal in terms of vectors and vector space?

I am beginner to linear algebra. I want to know detailed explanation of what is the difference between these two and geometrically how these two are interpreted?
0
votes
0answers
8 views

Positive hyperplane? Is there a name for these type of hyperplanes?

Consider an affine hyperplane $\{ x : \langle x,v\rangle=a \}$ where $a\ge 0$ and $v\in\mathbb{R}^n_{+}$. That is, both the level $a$ and the vector $v$ are non-negative. Is there any special name ...
0
votes
3answers
26 views

How can I show that and $n\times{n}$ matrix of the form in the description has a determinant of zero for $n>2$?

In General, $n>2$, $a_{i,j}=a_{i,j-1}+1$ and the matrix will be of the following form: ...
0
votes
1answer
23 views

Prove whether the linear equations are solvable or not?

I am beginner to linear algebra. I am confused for finding the solution for following question. There are set of linear equation(m equations and n unknown) represented in the form of matrices. ...
1
vote
2answers
27 views

Orthonormality of the columns of a matrix

I am studying orthogonal columns and matrices right now and I have encountered the following theorem: Theorem An $m \times n$ matrix $U$ has orthonormal columns if and only if $U^T U = 1$. Is it ...
1
vote
0answers
14 views

How to determine positive or negative definite of a bordered Hessian ?

I want to determine the minimization result I get using Lagrange Multiplier method is a local minimum by determining whether the Bordered Hessian is positive definite or negative definite.(Hopefully ...
0
votes
1answer
19 views

Help on Solutions to Systems of Equations

Here is a screenshot: http://imgur.com/gallery/Wh6ksgO/new I was looking at my Linear Algebra quiz solutions and I saw the following: "Thus from RREF, we can see the system if consistent and contains ...
0
votes
1answer
18 views

Given line $e$ and plane $\alpha$, find all points $Q$ on $e$ such $d(Q, P)= d(Q, \alpha)$

Can someone help me with this question and show my step by step process. I am unable to solve it. Thank you. $P(4,2,5)$ The plane $\alpha$ is given by $2x+y-2z=2$ The line $e$ is given by ...
2
votes
1answer
13 views

Commutation of a partial trace with an operator

Let the partial trace $\mathrm{tr}_B$ be a mapping from an endomorphism End$\left( H_A\otimes H_B \right)$ onto an endomorphism End$\left( H_A \right)$. Then the partial trace is defined as $$ ...
1
vote
2answers
38 views

What is switching rows useful for?

I've learned about elementary row operations, there is one of them that seems a little bit weird to me: The row switching. It seems that a system of equations: $$\begin{eqnarray*} ...
0
votes
0answers
19 views

Bound on the difference of matrix diagonals

I have two diagonal matrices $\Lambda,\hat{\Lambda}\in\mathbb{R}^{n\times n}$ with non-negative diagonal elements. And I have two matrices $W,\hat{W}\in\mathbb{R}^{m\times n}$, with $m\geq n$, each ...
0
votes
1answer
24 views

Learning to solve complex inequalities in many variables

below is a very specific inequality problem. I would like to know how to solve it so I can apply it to more complex problems. The equations are as follows: $$3.5x−2.5y−3z=A$$ $$−7.5x+3.75y+5.25z=B$$ ...
1
vote
0answers
13 views

matrix with fractional exponent, not getting expected output in Matlab/Octave

I have a matrix exponential function that is called a number of times in an integration routine from the heat conduction model I'm trying to implement. It works, and my results match the samples in ...
0
votes
0answers
12 views

Simulate ICA Source Signal

I am using the fastICA package in R for a matrix of time series information. However, if I wanted to simulate the process for risk management purposes how exactly could I do this? For example lets ...
2
votes
1answer
64 views

Find integer solutions equation of ${ x }_{ 1 }^{ 4 }+{ { x }_{ 2 }^{ 4 }+ }{ x }_{ 3 }^{ 4 }+…+{ x }_{ 14 }^{ 4 }=1599 $

I tried to solve this equation,but can't end up $${ x }_{ 1 }^{ 4 }+{ { x }_{ 2 }^{ 4 }+ }{ x }_{ 3 }^{ 4 }+...+{ x }_{ 14 }^{ 4 }=1599$$ My work: Consider arbitrary $x_{ i }=2k,\quad \forall ...
0
votes
0answers
15 views

System of linear equations and Fredholm's alternative

I am learning linear algebra and bought the book from Gilbert Strange: Introduction to linear algebra and trying to understand the four fundamental subspaces. I know that a system is solvable if b is ...
0
votes
1answer
19 views

A simple proof for angle inequality in inner product spaces

I am looking for a smiple proof for the following fact: Let $u,v,w$ be vectors in an inner product space $V$. Then it holds: $\theta (u, v)≤\theta(u, w) + \theta(w, v)$ (Of course if they are all in ...
3
votes
2answers
50 views

Finding a basis of a complex vector space over $\Bbb R$ given a basis over $\Bbb C$

Suppose $X$ is a vector space over $\mathbb C$ and has as basis $\{e_1,e_2,\ldots,e_n\}$. Now regard $X$ as a vector space over $\mathbb R$. What will be the basis? My thoughts: I considered ...
0
votes
0answers
9 views

Update PageRank given extra links

I have a stabilized importance vector $x_k$ that is the PageRank of a series of webpages as defined by the links between them. Graphically, this is the equivalent of a graph where nodes are pages and ...
2
votes
0answers
8 views

Lipschitz continuity of invariant subspaces for parametrized matrices

Let $A(t)$ be a one-dimensional parametrized family of linear operators on $\mathbb{R}^m$ that has smooth dependence on $t$. Let $V_0\subset \mathbb{R}^n$ be an $n$-dimensional invariant subspace for ...
1
vote
1answer
31 views

Given an area, calculate the angle of a wedge out of an annulus between a square and a circle

If we have a shape similar to this picture: Where the square length is less or equal to the circle's diameter, then I believe the term for the blue area is the annulus. I was wondering if it is ...
-1
votes
0answers
26 views

Signature of a complex matrix

If $A\in M_n(\Bbb R)$, we define its signature as the triple $(s^+,s^-,s^0)$, where $s^+,s^-,s^0\in\Bbb Z_{\ge0}$ denotes respectively the number of eigenvalues $>0,<0,=0$. How can we define ...
8
votes
1answer
50 views

Is an infinite system of (linear) equations solvable if all finite subsystems are?

I was wondering about the following. Let $A$ be an abelian group, $a_i$ variables indexed with some arbitrary set $I$ and assume we have an infinite set $E$ of linear equations in finitely many ...
1
vote
0answers
31 views

Transformation law for symmetric rank-2 tensors?

A rank-2 tensor $M_{ij}$ transforms as $M_{ij} \rightarrow O_{ik} O_{jl} M_{kl}$, where $O$ is some element of $SO(n)$. We can always get a symmetric tensor from $M_{ij}$ through $M_{ij}^s =M_{ij} + ...
0
votes
0answers
10 views

How to determine the minimum number of basis functions thats linear superposition best reproduces a set of curves?

How to determine the minimum number of basis functions that's linear superposition best reproduces a set of arbitrary curves?
0
votes
0answers
20 views

A question in matrix polynomial.

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
0
votes
0answers
21 views

Why do uniform, strong and weak convergence coincide for finite dimensional vector spaces?

For linear operators $A_n$, $A$ in a finite dimensional vector space $V$, I am trying to prove the equivalence of $\|A_n - A\| = \sup_{x \in \mathbb{C}^n, |x| = 1} |A_nx-Ax| \to 0$ as $n \to ...
2
votes
2answers
24 views

Matrix representation with non-standard bases.

In chaprer 2.2 of Fiedberg's Linear Algebra is wroten about matrix representation. But all examples are only with standard ordered bases. I made a task to understand it. Please, could you show me ...
5
votes
4answers
62 views

Given vector $\vec x = \left\{ x_i\right\}_{i=1}^n$ find an algebraic expression for $\vec y = \left\{ x^2_i\right\}_{i=1}^n$

Given vector $$\vec x = \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix},$$ How can we write out vector $$\vec y = \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix} := \begin{bmatrix} x^2_1 \\ ...
2
votes
2answers
15 views

Closest point of parameterized curve has orthogonal position vector to tangent

Let $\alpha(t)$ be a parameterized curve which does not pass through the origin. If $\alpha(t_0)$ is a point of the trace of $\alpha$ closest to the origin and $\alpha'(t_0)\ne 0$, show that the ...
-1
votes
3answers
120 views

$A$ is a $n \times n$ matrix over $\mathbb{R}$ such that $A^2+A+5I=0$. Find the characteristic polynomial of the matrix $A$. [on hold]

$A$ is a $n \times n$ matrix over $\mathbb{R}$ such that $A^2+A+5I=0$. Find the characteristic polynomial of the matrix $A$. Any ideas?
1
vote
0answers
27 views

Derivations of important algebras?

After knowing the importance of studying derivations of an algebras(Why we wonder to know all derivations of an algebra?, this problem naturally raised "what is the space of all derivations of ...
0
votes
3answers
34 views

Finding the kernel of a linear map

Our exercise is to find all solutions to the equation $Ax = 0$, among others for the following matrix $$A =\begin{pmatrix} 6 & 3 & -9 \\ 2 & 1 & -3 \\ -4 & -2 & 6 ...
2
votes
1answer
25 views

Perpendicularity in matrix space

Let $K$ and $Q$ be symmetric real matrices such that $K+Q$ is positive semidefinite ($\ge0$). My question is two questions: Does $KQ=0$ imply $K\ge0$ and $Q\ge0$? Does trace$(KQ)=0$ imply $K\ge0$ ...
-1
votes
0answers
28 views

Find the adjoint of the Linear OperatorT [on hold]

Find the adjoint of the Linear Operator $T:\mathbb R^3 \rightarrow \mathbb R^3$ defined BY $T(x,y,z) = (x+2y,3x-4z,y)$
1
vote
1answer
13 views

The projection $EF=E$ imply $M_2\subset M_1$?

Suppose $F$ is a projection on $M_1$ along $N_1$, $E$ is a projection on $M_2$ along $N_2$, if $EF=E$, does that imply that $M_2\subset M_1$?
0
votes
0answers
6 views

Is the basis for $\mathbb{R}^n$ where the $\mathfrak{so}(n)$ Cartan elements are diagonal, necessarily complex?

This is a follow-up to this and this question. The elements of $\mathfrak{so}(n)$ are antisymmetric in the standard basis $(1, 0, 0), (0, 1, 0), (0, 0, 1)$. This means that we have no diagonal ...
2
votes
1answer
25 views

$U^TA_1V$ is a rank-one matrix?

To give a little bit of context, the question I am asking is related to SVD decomposition. More specifically, we are trying to prove that the best rank one approximation for $A_1$ is $\sigma_1 u_{1} ...
0
votes
0answers
19 views

Can we express any matrix as an outer product expansion?

Suppose $XY$ is an $m $ by $n$ matrix, where $X$ is a $m$ by $k$ matrix and $Y$ is a $k$ by $n$ matrix. $y_i$ are the columns of $Y$ and $x_i$ are the columns of $X$. How do we know that ...
0
votes
1answer
40 views

Quadratic forms and midpoints

The midpoint of the vectors $u$ and $v$ is $w=\frac{u+v}{2}$. In euclidean geometry, an alternative characteristic of midpoints is $|v-w|=|u-w|=\frac{1}{2}|u-v|$. I wonder if this generalizes to ...
0
votes
1answer
33 views

More on linear algebra vector subspaces

I am continuing on my journey of trying to understand vector subspaces. Question: Let $F(-\infty,\infty)$ be the set of all real-value functions defined at each x in the interval $(-\infty,\infty)$. ...
-3
votes
0answers
42 views

What do accountant's learn? [on hold]

Since a high school student can find compound interest, calculate stock yields, etc. what does an accountant learn in college? Is it just busy work?
1
vote
2answers
24 views

Closed under scalar multiplication [on hold]

The subset of $\mathbb{R}^2$: $\{ (x,y)| y=\frac{7}{2}x\}$ is a subspace of $\mathbb{R}$. How can I prove that the subspace is nontrival ?
2
votes
1answer
49 views

Diagonalization and Commuting Matrices

Attempt: I have shown part (ii) and I have found the eigenvalues and eigenvectors for A, B respectively and shown they can be diagonalised. I need help with (iii) and (iv), for (iii) I can't show ...
0
votes
2answers
33 views

How to find $f$ for a symmetric bilinear form?

Let's say we have the symmetric matrix:$$A = \left(\begin{array}{cc} 1&2 \\ 2&0 \end{array}\right)$$ How do I find the symmetric bilinear form of this $A$?
-5
votes
1answer
42 views

What is the fastest, most correct way to solve this simultaneous of two linears?

\begin{eqnarray*} (x+2)/5-((y+2)/4) &=& 2-(x/3) \\ (x+5)/4+((x-y)/5) &=& y+5 \end{eqnarray*} What is the fastest, most correct way to solve this simultaneous of two linears?
-1
votes
6answers
82 views

Find $x$ and $y$ - Why is there no answer?

I need to find $x$ and $y$ from the following equations: \begin{eqnarray*} 7x-3y &=& 8 \\ 14x-6y &=& 21 \end{eqnarray*} I my book it says there's "no answer". Can someone explain to ...