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 ...

learn more… | top users | synonyms

0
votes
0answers
4 views

Quadratic form in canonical form relation

The homogeneous quadratic equation can be written as a matrix. It is also written as a canonical form by using orthogonal transformation. Why we are going for canonical form and what is the relation ...
12
votes
1answer
316 views

How to prove this determinant is $\pi$?

prove or disprove $$\pi=\begin{vmatrix} 3&1&0&0&0&\cdots\\ -1&6&1&0&0&\cdots\\ 0&-1&\dfrac{6}{3^2}&1&0&\cdots\\ ...
1
vote
0answers
29 views

Calculating Vandermonde determinant

I understand that the Vandermonde determinant $$ W(x_1, \ldots, x_n) = \left| \begin{array}{cccc} 1 & 1 & \cdots & 1\\ x_1 & x_2 & \cdots & x_n \\ x_1^2 & x_2^2 & ...
0
votes
3answers
761 views

How to show that $\{t, \sin t , \cos 2 t , \sin t \cos t \}$ is a linearly independent set of functions on $\mathbb{R}$?

I have this homework question that I have no idea how to do: Show that $\{t, \sin(t), \cos(2t), \sin(t)\cos(t) \}$ is a linearly independent set of functions defined on $\mathbb{R}$. Start by ...
2
votes
1answer
59 views

Proof $p(A)=0$ without Cayley-Hamilton theorem when $A$ is upper triangular

I need help proving $p(A)=0$ without Cayley-Hamilton theorem when $A$ is upper triangular, as part of the proof of the Cayley-Hamilton theorem The result makes a lot of sense but I can't prove it ...
0
votes
1answer
20 views

How do you find the vector x determined by the given coordinate vector and given basis B?

I saw a couple different ways to approach this problem from tutorials on YouTube, and each led to a different answer. This is what I got: 3 -4 | 5 -5 6 | 3 3 * 5 + -4 * 3 ...
0
votes
1answer
35 views
+50

Rotate secondary vanishing points to the primary vanishing points to find new length of object

all though only the 2D data is available, the best way to think of this problem is a piece of paper pinned at one corner to a wall, but the paper is sitting at an angle to the wall, see illustration ...
1
vote
1answer
43 views

When and why can functions “take on” the role of vectors in defining vector speaces?

In what I call "advanced" linear algebra, we examine the properties of vectors in a vector space like an inner product space by checking that they satisfy e.g. the Cauchy-Schwartz inequality, the ...
0
votes
0answers
105 views

Can we prove a connection between sum of numbers and $L_2$-like norm?

Let $u$ be a fixed vector of length $K$ and $A$ be a matrix of all positive numbers of size $K \times K$. Let $V$ be a set of vectors. Let $V(\epsilon) \subseteq V$ be a set of vectors of length $K$ ...
2
votes
0answers
49 views

Inverting the infinite matrix with entries $\mathbf{P}_{ij}={i-1\choose j-1}$

Let $ \mathbf{P}$ denote the "infinite matrix" $$ \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & \dots \\ 1 & 1 & 0 & 0 & \dots \\ 1 & 2 & 1 & 0 & \dots ...
6
votes
2answers
149 views
+50

Eigenvalue test faster than $O\left(n^3\right)$?

Given a real $n\times n$ matrix $A$, one can find the eigenvalues in $O\left(n^3\right)$ by using say, the $QR$ algorithm. Now, what if we guess an eigenvalue $\lambda_0$, and we want to know if it's ...
3
votes
2answers
252 views

Problems on Symmetric Matrices

1 . Let $A = (a_{ij})$ be a real $n \times n$ matrix such that $a_{ij} = a_{ji}$ for all $1 \leq i,j \leq n$ and $a_{ij} = 0$ for $|i-j|>1$. Moreover $a_{ij}$ is non-zero for all $i$,$j$ satisfying ...
2
votes
2answers
71 views

Prove there is a subspace of $V$ isomorphic to $T(V)$

If $T:V\to V$ is a linear transformation and $T(V)$ is of finite dimension then prove that there is a subspace $U$ of $V$ isomorphic to $T(V)$ and then show that, if $x,y\in V$, then $(x+U)\cap ...
0
votes
2answers
34 views

The set consisting of all solutions of a homogeneous linear differential equation of order $n$ is a vector space.

The set $S$ consisting of all solutions of a homogeneous linear differential equation of order $n$ is a vector space.
1
vote
4answers
146 views

How to deal with linear recurrence that it's characteristic polynomial has multiple roots?

example , $$ a_n=6a_{n-1}-9a_{n-2},a_0=0,a_1=1 $$ what is the $a_n$? In fact, I want to know there are any way to deal with this situation.
0
votes
1answer
21 views

Describe the solution set of the system

Consider the linear system below: $$\begin{array}{ccccccc} x_1&-&2x_2&+&&&x_4&=&1\\ 2x_1& -& 5x_2& -& 2x_3& +& k^2x_4 &= &-2\\ ...
1
vote
2answers
66 views

Prove that $A \circ B = AB$ if and only if both $A$ and $B$ are diagonal

Definition. Hadamard product. Let $A,B \in \mathbb{C}^{m \times n}$. The Hadamard product of $A$ and $B$ is defined by $[A \circ B]_{ij} = [A]_{ij}[B]_{ij}$ for all $i = 1, \dots, m$, $j = 1, \dots, ...
2
votes
4answers
52 views

Show that $ax^2+2hxy+by^2$ is positive definite when $h^2<ab$

The question asks to "show that the condition for $P(x,y)=ax^2+2hxy+by^2$ ($a$,$b$ and $h$ not all zero) to be positive definite is that $h^2<ab$, and that $P(x,y)$ has the same sign as $a$." Now ...
3
votes
1answer
39 views

Rank of the product of 3 matrices

Suppose I have 3 n by n matrices $A,B,C$ with $ABC=0$, what could be the maximal rank of $CBA$? I guess the answer would be n but I failed to prove it( tried to use Rank-Nuillity Theorem but I don't ...
3
votes
1answer
86 views

Changing local coordinates on a manifold by a diffeomorphism

This is the set up of my problem: Let $M$ be a manifold with local coordinates $x^1,\dots, x^n$. Let $x^1,\dots,x^n,\xi_1,\dots,\xi_n$ denote the induced coordinates on $T^\ast M$. Let $f:M\to M$ be ...
0
votes
1answer
54 views

Geometric meaning of line equation in homogeneous coordinate

In Euclidean space, a line's equation is $$ax + by + c = 0.$$ While in homogeneous coordinates,it can be represented with $$\begin{pmatrix}x &y &1\end{pmatrix}\begin{pmatrix}a\\ b\\ ...
4
votes
2answers
2k views

Span of an empty set is the zero vector

I am reading Nering's book on Linear Algebra and in the section on vector spaces he makes the comment, "We also agree that the empty set spans the set consisting of the zero vector alone". Is Nering ...
0
votes
1answer
28 views

Matrix Multiplication - When do you only multiply by one number and add vs. multiplying all numbers?

*I wasn't sure where to put this. Just let me know if I should delete it or if there is another category/website where this question would fit better. Thanks! Or if you know the answer & don't ...
0
votes
1answer
27 views

Proving $\mathrm{Hom}(V \rightarrow W)$ is a vector space

It can easily be proven that $\newcommand{\Hom}{\mathrm{Hom}}\Hom(V \rightarrow W)$ is a sub-space. 1. we know that for any $T:V\rightarrow W$, T(0)=0, therefore $0\in \Hom(V \rightarrow W)$ 2. ...
1
vote
1answer
54 views

Is it true that $d\textbf{S} = dy dz\textbf{ i }+ dx dz\textbf{ j }+ dx dy\textbf{ k }$

I came up with this in my mind, Just wondering if it is true I am thinking about it too, will post my observations, if any
2
votes
0answers
36 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 ...
1
vote
1answer
35 views

Hamming Code (9,5): Is my Parity Check correct?

I have an exercise about designing parity checks for the Hamming (9,5) group code with minimum distance $3$. I use the following notation for the generator matrix: $$ ...
1
vote
1answer
924 views

How I can get coefficients matrix in MATLAB from the set of linear equations?

I am working in MATLAB and having problem in extraction of coefficients out of my set of symbolic linear equations. Equations are as such R = [0.8978*c - 0.8047*b - 0.020299*a, 0.8978*d - ...
3
votes
2answers
26 views

problem with denominator in transformation

hi i cant understand where the 2 comes from in this transformation any help would be appreciateD
2
votes
3answers
45 views

Determinant-like expression for non-square matrices

I'm interested in whether for any real matrix of size $m \times n$ there is a real number with the following properties: It is a polynomial expression with real coefficients in the entries of the ...
0
votes
1answer
36 views

calculus / algebra

Hi can anyone go through the transformation of the equation below as i cannot understand where the 2 in comes from any help would be much appreciated $$\frac{\omega k^{0.5}}{\omega k} = ...
-1
votes
0answers
19 views

Positive definite [on hold]

I need a graphical representation of positive definite from the eigen values of the matrix which can be expressed from the second degree homogenous equation.
1
vote
0answers
14 views

Polynomial Discriminant of 4*g+h^2

Suppose we have two polynomials $g,h\in \mathbb Z[x]$ with $\deg g = 2k+1 =:n$ and $\deg h=k$. As an example, take $g=x^7+2\cdot x^6+x+2$, $h=x^3+x+7$. My question is: Why does the discriminant of ...
2
votes
1answer
22 views

Determinant of a rank-one update of a scalar matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few): Eigenvalues of a matrix of $1$'s ...
0
votes
0answers
40 views

Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
1
vote
1answer
28 views

Positive definite matrix.

How to illustrate the positive definite matrix in vector space by using the eigen values and eigen vectors?
1
vote
1answer
48 views

Matrix of rank one.

Let $A\in M_n(\mathbb R)$ be a matrix of rank $1$. Show that $A$ is similar to a matrix of the form $$A'=\begin{pmatrix} 0 & \cdots &0& a_1 \\ 0& \cdots&0& a_2\\ \vdots ...
0
votes
0answers
37 views

Subset of vector space containing zero vector. [on hold]

Subset of vector space containing zero vector is always linearly independent.Is this statement is true?
1
vote
4answers
102 views

Finding the characteristic polynomial of this specific $3\times3$ matrix

How can I find the characteristic polynomial of the following matrix: \begin{pmatrix} 0&-2&2\\-2&1&0\\2&0&-1 \end{pmatrix} please I need the details.
0
votes
2answers
32 views

Proof that Every Positive Operator on V has a Unique Positive Square Root

Suppose V is a finite-dimensional, nonzero, inner-product space over F, and F denotes R or C. My thought is : suppose T is a positive operator; thus, T is self-adjoint. Every self-adjoint operator on ...
-2
votes
0answers
17 views

Form the biquadratic equation two of whose roots are i and 3 . [on hold]

plese answer this quikly Form the biquadratic equation two of whose roots are i and 3 .
-3
votes
0answers
13 views

T has zero as a characteristic root [on hold]

Let $ V $ be a vector space over field $F$. $T$:$V$$\rightarrow$$V$linear transformation such that $T$ has zero as a characteristic root.Then 1.T is diagonalisable over $F$ 2.Multiplicity of each ...
8
votes
0answers
89 views
+50

Families of Idempotent $3\times 3$ Matrices

I did the following analysis for $2\times2$ real idempotent (i.e. $A^2=A$) matrices: $$ ...
3
votes
2answers
52 views

Is calling a linear-equation a linear-function, misnomer or completely wrong?

From my college life, I remember many professors used to call a linear-equation a linear-function, however: A standard definition of linear function (or linear map) is: $$f(x+y)=f(x)+f(y),$$ ...
23
votes
7answers
2k views

Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the ...
3
votes
0answers
22 views

Reference for Generalized Eigenvectors

I am looking for references on generalized eigenvectors and Jordan matrix representation. I would like a brief but complete introduction of this concepts with a nice treatment of the most important ...
0
votes
1answer
22 views

Identifying if a $ S $ given is a vector subspace

Could you help me to identify if $ S $ is a vector subspace? I started learning linear algebra and I get this question and I am lost.
0
votes
0answers
38 views

Show that vectors of the form (a,b,1) do not form a vector space

Show that vectors of the form $(a,b,1)$ do not form a vector space I tried using the vector space axioms to attack the problem but I am not going anywhere with this problem. I do not need a ...
5
votes
5answers
628 views

Prove that if Rank$(A)=n$, then Rank$(AB)=$Rank$(B)$

Let $A \in Mat_{m\times n}(\mathbb{R})$ and $B \in Mat_{n\times p}(\mathbb{R})$. Prove that if Rank$(A)=n$ then Rank$(AB)=$Rank$(B)$. I tried to start with definitions finding that $n \le ...
4
votes
1answer
166 views

What is the connection between $\rho$ and $\sigma$ if $\rho\rho^T=\sigma\sigma^T$?

I want to prove that there exists a Borel function $R(\rho,\sigma)$ with values in $M^{d\times d}$ defined on $D=\lbrace(\rho,\sigma)\in M^{d\times d}\times M^{d\times d}\,: ...