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 ...

learn more… | top users | synonyms

1
vote
2answers
58 views

Calculating time to 0?

Quick question format: Let $a_n$ be the sequence given by the rule: $$a_0=k,a_{n+1}=\alpha a_n−\beta$$ Find a closed form for $a_n$. Long question format: If I have a starting value $x=100000$ ...
1
vote
1answer
2k views

Combine transformation matrices

Question: Find the transformation matrix that combines the following transformation matrices, in order: $$\begin{bmatrix} &3 &0 &0 &0 \\ &0 &-1 &0 &0 \\ ...
2
votes
2answers
71 views

Showing $(v - \hat{v})\,\bot\,v$

$\fbox{Setting}$ Let $V$ be an inner-product space with $v \in V$. Suppose that $\mathcal{O} = \{u_1, \ldots, u_n\}$ forms an orthonormal basis of $V$. Let $\hat{v} = \left\langle u_1, ...
4
votes
1answer
81 views

Revisted: $T^{**}\circ \varphi_1 = \varphi_2\circ T$

So, I'm trying to show that $T^{**}\circ \varphi_1 = \varphi_2\circ T$ where $\varphi_1 : T\rightarrow V^{**}$ and $\varphi_2 : W\rightarrow W^{**}$. Also, $T^{**} : V^{**}\rightarrow W^{**}$, and yes ...
3
votes
3answers
80 views

How to construct a matrix $A$

Construct a matrix $A$ such that $A^2\ne 0$ but $A^3=0$. I need your help to find $A$. Please help. Thanks in advance.
0
votes
0answers
70 views

Generalizing formula for calculating determinant of specific matrix

There is a similar question like this. And this is extension of this question How can we calculate the determinant of this $\,pn-1\times pn-1\,$ matrix. I have tried at my best level, and still am ...
1
vote
1answer
249 views

Show that the zero set of $f$ is an orientable submanifold of $\Bbb R^{n+1}$.

Suppose $f(x_1,...,x_{n+1})$ is a$ C^∞$ function on $\Bbb R^{n+1}$ with $0$ as a regular value. Show that the zero set of $f$ is an orientable submanifold of $\Bbb R_{n+1}$. In particular, the unit ...
2
votes
2answers
95 views

Simultaneously solving of equations

I am trying to refresh some math skills and I am struggling over the following problem. I tried to solve it with the help of a number of sources (i.e. http://www.idomaths.com/simeq.php), but I haven't ...
2
votes
2answers
76 views

Dimension of the sum of subespaces

Let $V_1$ and $V_2$ be two subspaces of a vector space of finite-dimension, such that $$\mbox{dim}(V_1+V_2)\ =\ \mbox{dim}(V_1\cap V_2) + 1,$$ show that $V_1 \subseteq V_2$ or $V_2 \subseteq V_1$. ...
1
vote
1answer
356 views

How to decide whether F is orientation-preserving or orientation-reversing as a diffeomorphism onto its image.

Let $U$ be the open set $(0,∞)×(0,2π)$ in the $(r,θ)$ -plane $R^2$. We define $F : U ⊂ R^2 → R^2$ by $F (r, θ ) = (r cos θ , r sin θ )$. How to decide whether F is orientation-preserving or ...
7
votes
1answer
189 views

Basis for $\mathbb{[Q(\pi):Q]}$

I'm trying to figure out whether the basis of $\mathbb{Q}(x)$ over $\mathbb{Q}$ is countable when $x$ is transcendental. I know that the elements in $\mathbb{Q}(x)$ will be rational functions in $x$ ...
1
vote
1answer
119 views

All the logarithms of a non-singular matrix.

I'm reading some notes on dynamical systems that talk about matrix logarithms with little to no detail on the subject. I read the wikipedia article and others on the internet, but not all is clear. ...
1
vote
1answer
34 views

Find a matrix A with $A \in M(6,4)$ with $\text{dim}(\text{ker}(A))=2$

Find a matrix A with $A \in M(6,4)$ with $\text{dim}(\text{ker}(A))=2$ Because $\text{dim}(\text{ker}(A)) + \text{dim}(\text{rg}(A))=$number of columns $\rightarrow 2 + \text{dim}(\text{rg}(A)) = 4$ ...
6
votes
1answer
202 views

“Convex” polynomials

Let me define "convex" polynomials, as the smallest class $\mathcal{C}$ of functions $p:\mathbb{R}\rightarrow \mathbb{R}$ defined (inductively) as: UPDATED (case 0 was missing): 0) $p(x)=x$, i.e., ...
3
votes
2answers
631 views

Find the rank of the following matrix.

$A= \left[ \begin{array}{ccc} 3 & -1 & 2 \\ -6 & 2 & 4 \\ -3 & 1 & 2 \end{array} \right]$ Applying, $R_{3}-\frac{1}{2}R_{2}$ ~ $A= \left[ \begin{array}{ccc} 3 & -1 & ...
0
votes
1answer
28 views

proper interpretation for these notations

we have this 2 equations listed below eq 1. δ[n] = u[n] - u[n-1] eq 2. y[n] = x[n] - x[n+1] Now the question is, 1.) which of the above equation corresponds to a ...
1
vote
1answer
104 views

A problem on calculating rank of a matrix

Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a $4 \times 4$ matrix A by $$\left( \begin{array}{ccc} x_1^2+y_1^2 & ...
1
vote
0answers
346 views

Find the position vector of the point R that is closest to the origin on the plane a'x + b'y + c'z = e

(a) Write down the Cartesian equation for the plane through the point Q(1,0,0) with normal n = -$ \sqrt{6} $ i - $ \sqrt{2} $ j - k and compute the distance of the origin from this plane. (b) Let ax ...
0
votes
1answer
86 views

Inverse of a transformation

How can one show that the inverse of the transformation $$x=y+h(y)$$ $x$,$y$ in $n$ dimensional real space and $h$ is an $r$-th degree homogeneous polynomial in $y$ has an inverse at $0$ in $n$ ...
0
votes
3answers
60 views

Inverting a system of equations

$$a_{out} = a\ \mathrm{\cos}(\theta) + i b\ \mathrm{\sin}(\theta)$$ $$b_{out} = b\ \mathrm{\cos}(\theta) + i a\ \mathrm{\sin}(\theta)$$ What procedure would one use to invert this to get: $$a = ...
0
votes
3answers
48 views

Question on Linear Transformation

Suppose $ T:\mathbb R^2 \rightarrow \mathbb R $ is a linear transformation such that the kernel of T is {$(x,-x): x\in\mathbb R $}. If T takes (1,0) to 1 then T takes (1,1) to which number? I tried ...
1
vote
2answers
52 views

Pointwise order on polynomials

I'm curious about the following problem. Given two polynomials $p,q:\mathbb{R}\rightarrow \mathbb{R}$ is it possible to determine automatically if $p(x)\leq q(x)$ for all $x\in\mathbb{R}$? I assume, ...
0
votes
1answer
213 views

find the projection of a vector on a line using the projection formula

In the following problem: "Find the projection of vector $\vec{v} = (2, 3)$` onto the line $y = 2x -1$" What are the beginning steps in solving this problem ? Do I pick a point on the line and then ...
2
votes
2answers
469 views

Exploring underdetermined linear system with non-negative solution

I haven't had much luck searching for this specific problems. Any pointers would be greatly appreciated. I have an underdetermined system where $ A $ and $ b $ are known. $ x $ is a real vector with ...
2
votes
1answer
196 views

Solving linear equations with Vandermonde

Given this: $$\begin{pmatrix} 1 & 1 & 1 & ... & 1 \\ a_1 & a_2 & a_3 & ... & a_n \\ a_1^2 & a_2^2 & a_3^2 & ... & a_n^2 \\ \vdots & \vdots & ...
1
vote
2answers
61 views

Is the matrix V in the subspace U?

I'm given that $U$ is the subspace of $M(3,2)$ generated by $A=\begin{bmatrix} 0 & 0 \\ 1 & 1 \\ 0 & 0 \end{bmatrix}$, $B=\begin{bmatrix} 0 & 1 \\ 0 & -1 \\ 1 & 0 ...
3
votes
3answers
148 views

Inverse of a symmetric tridiagonal filter matrix

How to get the inverse of this matrix: $\left(\begin{array}{ccccccc} ...
0
votes
1answer
148 views

Closeness of any matrix to a diagonalizable matrix in terms of norm-2

Assuming $X \in \mathbb{C}^{n \times n}$, how to show that for all $\epsilon > 0$, there exist a diagonalizable matrix $D \in \mathbb{C}^{n \times n}$ such that $\left \| X-D \right \|_2 < ...
2
votes
0answers
152 views

Transposition of Composition is Reversed Composition of Transpositions

I'm trying to show that $(UT)^*=T^*U^*$. Here is my effort: Consider the following data: \begin{array}{lcl} T:V\rightarrow W & \leadsto & T^*:W^*\rightarrow V^* \\ U:W\rightarrow Z & ...
2
votes
1answer
153 views

Proving a diagonal matrix exists for linear operators with complemented invariant subspaces

I came across this problem one of my practice worksheets and I was stumped as to how I would go about solving this. Let $T : V \rightarrow V$ be a linear operator on a finite dimensional vector space ...
17
votes
1answer
814 views

Properties of the Cone of Positive Semidefinite Matrices

The set of positive semidefinite symmetric real matrices form a cone. We can define an order over the set of matrices by saying $X\geq Y$ if and only if $X-Y$ is positive semidefinite. I suspect that ...
1
vote
1answer
54 views

If basis $\beta$ is orthonormal, then $\beta^{*}=\beta$

Let $V$ be a finite inner product space. Suppose that $\beta$ is orthonormal basis of $V$. How do I show that $\beta^{*}=\beta$? Where $\beta^{*}$ is dual basis of $\beta$ Dual basis definition: ...
2
votes
1answer
674 views

Can infinite-dimensional vector spaces be decomposed into direct sum of its subspaces?

I'm reading Axler "Linear agebra done right" and in Chapter 1 he discusses subspaces and direct sum. My question is, are there subspaces of the infinite-dimensional vector spaces, e.g. a functional ...
3
votes
1answer
204 views

Rank of a block-triagonal matrix

Given a matrix $C=\left [ \begin{matrix} A & 0 \\ B & A \end{matrix} \right ]$, where rank(A+B)=rank(B), and rank(B)>rank(A), does rank(C)=rank(A)+rank(B) hold? A,B are Laplacian matrices.
0
votes
2answers
60 views

Column space and row space of a matrix of which $\det(A)=0$

If $\det(A)$ not equal $0$, then $\operatorname{Col} A = \operatorname{Row}A$? $A$ is diagonalizable? Thank you a lot.
1
vote
1answer
429 views

Dual basis existence and uniqueness.

In Wikipedia, on Dual Basis they say: "Algebraically, a dual set always exists, and gives an injection from $V$ into $V^*$. However, a dual basis exists if and only if a vector space is finite ...
1
vote
1answer
1k views

Writing a polynomial as a linear combination of other polynomials

I'm currently working on writing $3(x)_4 - 12(x)_3 + 4(x)_1 - 17$ as a linear combination of $(x)_4,\ldots,(x)_0$ and am having difficulty understanding where the conversion comes from. I have the ...
1
vote
2answers
47 views

Solution of system of linearly dependent equations.

So, I have the system of equations $x'(t) = Ax$ where $A$ is first row-(4,-2) and second row - (8,-4). This has two eigenvalues, both are 0. But I tried to solve it this way: $x_1' = 4x_1 -2x_2$ and ...
0
votes
0answers
49 views

Prove/disprove: if $d \mid f$ and $d \nmid g$ then we can not know if $d\mid (f+g)$ or $d \nmid (f+g)$

Given three polynomials $f,g,d \in \mathbb F[x]$, we need to prove or disprove the following assumption: if $d \mid f$ and $d \nmid g$ so we can not say for sure if $d \mid (f+g)$ or $d \nmid ...
3
votes
2answers
872 views

How to interpret “rank” of a matrix intuitively?

What is the physical interpretation of "rank" of a matrix ? Why is it called "rank" ?
1
vote
0answers
77 views

Vandermonde question

I'm studying time series analysis and in my book I came a cross with the following proof (The proof is actually the last page, but I posted as much information as possible on the problem): I have ...
0
votes
1answer
46 views

How do I show that $F^{∗}(dx∧dy∧dz) = ρ^{2} \sin φ dρ∧dφ∧dθ$.

I dont know how to solve. Please help me. I need to understand such types of the question for my exam studyings.
2
votes
1answer
2k views

Intuitive proof of row rank = column rank? [duplicate]

Is it possible to give an intuitive/elementary proof of the theorem that says that the row rank of a (finite-dimensional) square matrix matrix equals its column rank?
1
vote
2answers
44 views

Matrix Algebra Manipulation

How does one show with full calculations: $$S=\frac{1}{n}\sum_{i=1}^n(x_i-\bar x)(x_i-\bar x)^T = \frac{1}{n}\left(\sum_{i=1}^nx_ix_i^T\right) - \bar x\bar x^T$$ where $$\bar ...
2
votes
1answer
174 views

Determinant of matrix?

How can we calculate the determinant of this $\,pn\times pn\,$ matrix. I have tried at my best level, and still am not able to come up with a solution. The matrix $a_{ij}$ entry is defined as $$ ...
0
votes
4answers
293 views

Divide and Conquer matrices to calculate determinant.

Do the determinant of a matrix equal to the determinant of submatrices? $$ det\begin{pmatrix} a_{11} & a_{12} & a_{13} & \dots & a_{1k} \\ a_{21} & a_{22} & a_{23} & ...
1
vote
1answer
96 views

Questions about orthogonal matrices.

Let $a_1, b_1, a_2, b_2$ be vectors in $V$ with dimension $n$. Suppose that the lengths of $a_1, b_1$ are the same and the lengths of $a_2, b_2$ are the same. Suppose that the angle of $a_1, a_2$ is ...
0
votes
2answers
67 views

Converting orientation and speed to position

I have a body located at $(x,y,z)$ at time $t_0$. I know the body is moving at a constant speed, $s$ (I don't know the direction he's moving only the magnitude of the velocity vector). The body's ...
1
vote
1answer
157 views

The matrix of rotation

Melvin Schwartz starts Principles of Electrodynamics by matrix of rotations, on page 4: Wikipedia says something similar but less thoroughly, so I will not discuss it. Basically, page says that ...
1
vote
1answer
255 views

Proving $(\operatorname{ker}T)^{\perp}\subseteq \operatorname{Im} T^{*}$

Let V be a finite inner product space with $T:V\to V$ a linear transformation. How can I prove that, $(\operatorname{ker}T)^{\perp}\subseteq \operatorname{Im}T^{*}$ ? Edit: My purpose is to ...