Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, Hamel basis, dimension, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization...
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votes
0answers
8 views
$W^0$ \overset{?}{\text{is}} a subspace of $V^*$
If $W\subset V$ is a subspace of $V$, and $W^0=\{f\in V^* | f(v)=0~\forall~v\in W\}$, then how do I show that $W^0$ is a subspace of $V^*$?
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votes
0answers
7 views
Linear Algebra — Block Matrix Inversion
Please excuse my formatting...
$X=\left(\matrix{A & B\\C & D}\right)$ where $A,B,C,D$ are all $n\times n$ matrices. Assuming that all stated inverses exist show that ...
0
votes
2answers
21 views
Spanning a Vector space of matrices by symmetric and skew symmetric matrices.
How do I span a vector space of $4\times 4$ matrices with real values by symmetric and skew symmetric matrices?
The basis of vector space of $4\times 4$ matrices has 16 elements, each containing one ...
5
votes
1answer
40 views
A problem on linear algebra
If $M$ is a $3 \times 3$ matrix such that
$$[ 0 ~~1 ~~2 ]M = [ 1 ~~0~~ 0 ] \text{ and } [ 3~~ 4 ~~5 ]M = [ 0 ~~1 ~~0 ]$$
then what is the value of $[ 6 ~~7 ~~8 ]M$ ?
I guess some matrix property ...
0
votes
0answers
19 views
Relationship between decompositions of matrices with overlapping kernels
Suppose $A$ and $B$ are two positive semidefinite matrices of size $N\times N$, satisfying
$$
\ker(A) \subseteq ker(B) \quad\Rightarrow \ \ \textrm{Im}(B) \subseteq \textrm{Im}(A)
$$
Further assume ...
1
vote
1answer
19 views
Nilpotent matrices with same minimal polynomial and nullity
From Hoffman and Kunze.
Let $N_1$ and $N_2$ be 6 X 6 nilpotent matrices over the field F. Suppose that
$N_1$ and $N_2$ have the same minimal polynomial and the same nullity. Prove that
$N_1$ ...
2
votes
1answer
23 views
Ask a question about an example in a course note on optimization problem with equality constraint
I have two difficulties on understanding the solution to an example in a course I took this semester on optimization. This example is given to illustrate the usage of Lagrange multiplier method ...
4
votes
1answer
32 views
Calculate the determinant when the sum of odd rows $=$ the sum of even rows
I have came across this interesting question in linear algebra and I couldn't know for sure the answer.
Given a matrix $A \in M_{n \times k} (\mathbb F)$, The sum of odd rows of $A$ $=$ the sum of ...
1
vote
2answers
51 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 $x=100000$ ...
2
votes
2answers
55 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, ...
3
votes
1answer
46 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 ...
0
votes
1answer
11 views
How to solve an underdetermined linear system with variables limited to an interval
If I have an underdetermined linear system of equations, with the additional constraint that all of the variables are limited to the interval $[0, 1]$, what techniques are there to solve this in the ...
0
votes
0answers
30 views
Orientation-preserving diffeomorphisms
My question is that
I only know that an atlas on $M$ is said to be oriented if for any two overlap- ping charts $(U,x^1,...,x^n)$ and $(V,y^1,...,y^n$) of the atlas, the Jacobian determinant ...
0
votes
0answers
19 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 ...
0
votes
1answer
20 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
50 views
Simultaneously solving of equations
I am trying to refresh some math stills 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
39 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 $V_2 \subseteq V_1$.
...
0
votes
1answer
20 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 ...
6
votes
0answers
55 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 $x$ ...
0
votes
1answer
32 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.
...
0
votes
0answers
32 views
Linear Algebra and the determinant of matrix variation
Let's have matrix $\hat {M}$ and it's variation $\hat {M} + \delta \hat {M}$.
How to prove that
$$
det(\hat {M} + \delta \hat {M}) = det( \hat {M})\left( 1 + Tr(\hat {M}^{-1}\delta \hat {M}) + ...
1
vote
1answer
27 views
Find a matrix A with $A \in M(6,4)$ with $\text{dim}(\text{ker}(A))=2$
Find a matrix A with $A \in M(6,4)$ with $\text{dim}(\text{ker}(A))=2$
Because $\text{dim}(\text{ker}(A)) + \text{dim}(\text{rg}(A))=$number of columns
$\rightarrow 2 + \text{dim}(\text{rg}(A)) = 4$ ...
4
votes
0answers
43 views
“Convex” polynomials
Let me define "convex" polynomials, as the smallest class $\mathcal{C}$ of functions $p:\mathbb{R}\rightarrow \mathbb{R}$ defined (inductively) as:
UPDATED (case 0 was missing):
0) $p(x)=x$, i.e., ...
0
votes
1answer
18 views
proper interpretation for these notations
we have this 2 equations listed below
eq 1. δ[n] = u[n] - u[n-1]
eq 2. y[n] = x[n] - x[n+1]
Now the question is, 1.) which of the above equation corresponds to a ...
1
vote
1answer
37 views
A problem on calculating rank of a matrix
Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a $4 \times 4$ matrix A by
$$\left( \begin{array}{ccc}
x_1^2+y_1^2 & ...
1
vote
0answers
23 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 ...
1
vote
0answers
48 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
42 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:
$$a = ...
0
votes
3answers
32 views
Question on Linear Transformation
Suppose $ T:\mathbb R^2 \rightarrow \mathbb R $ is a linear transformation such that the kernel of T is {$(x,-x): x\in\mathbb R $}. If T takes (1,0) to 1 then T takes (1,1) to which number?
I tried ...
1
vote
2answers
31 views
Pointwise order on polynomials
I'm curious about the following problem. Given two polynomials $p,q:\mathbb{R}\rightarrow \mathbb{R}$ is it possible to determine automatically if $p(x)\leq q(x)$ for all $x\in\mathbb{R}$?
I assume, ...
0
votes
0answers
15 views
Proving a certain tridiagonal matrix is positive semi-definite
Start with $n+1$ positive numbers in increasing order, $\rho_1 \leq \rho_2 \leq ... \leq \rho_{n+1}$. Now define the symmetric tridiagonal matrix $M$ -
$$M=
\begin{bmatrix}
a_1 & -b_2 & ...
0
votes
1answer
17 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 ...
1
vote
1answer
25 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
37 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
29 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 ...
2
votes
2answers
51 views
Inverse of a symmetric tridiagonal filter matrix
How to get the inverse of this matrix:
$\left(\begin{array}{ccccccc}
...
0
votes
1answer
30 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 < ...
1
vote
0answers
38 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
24 views
Proving a diagonal matrix exists for linear operators with 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 ...
4
votes
0answers
27 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
28 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
34 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 ...
2
votes
1answer
148 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
0answers
20 views
Normal endomorphism
I have a question about normal endomorphism. In class, we said that normal endomorphisms in finite dimensional real vector spaces are always of the form that we have some eigenvalues and further ...
0
votes
2answers
42 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
35 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
39 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
3answers
32 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
29 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 ...
-1
votes
0answers
36 views
I wand to find the Proof of the pullback of a wedge product. Please let's suggest a book or website to find this proof.
Proposition :
If $F : N → M$ is a $C^∞$ map of manifolds and $ω$ and $τ$ are differential forms on M, then
$F^∗(ω ∧τ) = F^∗ω ∧F^∗τ$.
This is a proposition from my textbook. I am studying the exam. ...
