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, etc. For questions specifically concerning ...
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2answers
18 views
real numbers a vector space over rational numbers?
Let $V$ be set of real numbers and $K$ the field of rational numbers.
Is $V$ a vector space over $K$, with ordinary addition of real numbers
and multiplication by rational numbers?
3
votes
1answer
34 views
About $\mathcal{L}(V,W)$
Let $V,W$ are two vector space and let $S\subseteq V$. Define: $$S^{0}=\{T\in\mathcal{L}(V,W)\mid~T(x)=0, \forall x\in S\}$$ The problem aks me to verify $S^{0}$ is a subspace of $V$ and if $V_1,V_2$ ...
1
vote
1answer
14 views
Invariant hermitian forms and irreducible representations
Let $V$ be a vector space over $\mathbb{C}$ of finite dimension $n$, $G$ is a finite group and $T:G\rightarrow GL(V)$ its irreducible representation that sends each $g$ into $T_g$.
Let $E:V^{\bigoplus ...
0
votes
1answer
25 views
Minimum polynomial and matrix multiplication
May you help me with the following proving?
Let $A,B$ be square matrices over $\mathbb C$ and suppose that there exist rectangular matrices $P,Q$ over $\mathbb C$ such that $A=PQ$ and $B=QP$.
...
0
votes
1answer
33 views
Minimal polynomial of a linear operator
How to show the following:
Let $T: V_F \to V_F$ be a linear operator and $f(x)$ be the minimal polynomial of $T$ over $F$. Let $$f(x)=g_1(x)g_2(x)\cdots g_n(x)$$ where the $g_i$'s are monic and ...
1
vote
1answer
23 views
Composition of systems of equations
Suppose $$2x + 3y = u$$ $$x - 4y = v$$
and further that
$$3u - 5v = c$$ $$2u + 3v = d$$
Express c and d in terms of $x$ and $y$ by matrix multiplication.
It's quite easy by direct substitution but ...
0
votes
0answers
22 views
Using a matrix to organise values into groups
Let's say I have a matrix of size 6 x 6.
Six students are 'ranking' six other students (including themselves). If I wanted to organise them into let's say, groups of three without picking and ...
4
votes
4answers
55 views
Symmetric Matrices of $I_{2}$
Find 10 symmetric matrices $ A = \left| \begin{array}{cc}
a & b \\
c & d \\
\end{array} \right|$ such that $A^{2}=I_{2}$
(I'm going to call matrix A the "square root" of $A^{2}$. If this is ...
3
votes
2answers
30 views
Matrix multiplication related to complex numbers?
Evaluate and simplify the product
$\begin{bmatrix} r\cos(\alpha) & -r\sin(\alpha) \\ r\sin(\alpha) & r\cos(\alpha)\\ \end{bmatrix}$ $\begin{bmatrix} s\cos(\beta) & -s\sin(\beta) \\ ...
5
votes
1answer
48 views
Having trouble using eigenvectors to solve differential equations
The question asked to solve $$\frac{dx}{dy} = \begin{pmatrix}
5 & 4 \\
-1 & 1\\
\end{pmatrix}x$$ ,where $$ x = \begin{pmatrix} x_1 \\
x_2 \\ \end{pmatrix}$$
I went ...
3
votes
1answer
37 views
Does convex and radially open imply open?
I want to show that a convex set $A$ is radially open iff $A\cap W$ is open in W, for every finite dimensional linear subspace.
Here the 'openness' we are talking about is from any normed space.
...
4
votes
1answer
27 views
How to show that a valid inner product on V is defined with the formula $[x, y] = \langle Ax, Ay\rangle $?
Let $A \in L(V,W)$ be an injection and $W$ an inner product space with the inner product $\langle \cdot,\cdot\rangle $. Prove that a valid inner product on $V$ is defined with the formula $[x, y] = ...
1
vote
1answer
46 views
Groups of transformations
I tried to find literature and articles about groups of transformations, but mostly what I found is either about groups or about transformations.
Can you suggest me literature where groups of ...
2
votes
1answer
67 views
Given a vector space $V$, show that the following statements are equivalent.
Given a subset $W$ of $V$ then I want show that,
$\forall v \in V, w \in W$ $\exists \lambda \in \mathbb{R}$ such that $w + \alpha v \in W$ for any $0 < \alpha < \lambda$
iff
$\forall v \in ...
0
votes
1answer
24 views
how would $f(T)$ look like if …
I know the result that if $T:V_F\to V_F$ is a linear operator then for any polynomial $f(x)\in F[x],~f(T)$ is a linear operator. Now my question is how would $f(T)$ look like if
$f(x)$ is the zero ...
1
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1answer
41 views
Fast way to calculate Eigen of 2x2 matrix using a formula
I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html
Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
2
votes
1answer
33 views
calculate kernels of matrices with angles
So my professor gave me this question:
I have to find the basis of the eigenvalues of this matrix
\begin{pmatrix}
\cos(q) & \sin(q)\\
\sin(q) & -\cos(q)\\
\end{pmatrix}
so I calculate ...
4
votes
1answer
27 views
Consequences of a rectangular matrix being of maximal rank
I have a real matrix $A$, $(m+1) \times m$ and a vector $b \in \mathbb R^{m+1}$ such that $b_{m+1}=0$. For any vector $u\in \mathbb R^m$, $Au=0 \Rightarrow u=0$. This means that $A$ is a rectangular ...
1
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1answer
23 views
Some remarks/questions from Primary Decomposition Theorem to get verified
In course of self-studying the Canonical Form in Linear Algebra I'm trying to put some remarks from the concept I acquired from Primary Decomposition Theorem which reads as follows:
Let $T:V_F\to ...
0
votes
2answers
46 views
Cases where characteristic and minimum polynomial coincide
Given a matrix such as $$\pmatrix{0 & 0 & 2 \\ 1 & 0 & -1 \\0 & 1 & 1 \\ },$$ whose characteristic polynomial is $-X^3+X^2-X+2.$
$$$$How it could be deduced that it equals ...
0
votes
3answers
48 views
Vectors that form a triangle!
I have a problem here.
How can I prove that sum of vectors that form a triangle is equal to 0 (AB+BC+CA=0) ?
Thank you!
2
votes
1answer
45 views
Is the inverse function smooth?
Imagine that we have a function $Inv$ that maps $A \rightarrow A^{-1}$, where A is an invertible square matrix. now my questions is: how do i see that this function is arbitrarily often ...
3
votes
1answer
34 views
Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$
Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$
Using $\det(Q)\det(\bar{Q}^T) = I$, I get to the stage $\det(\bar{Q})\det(Q)=1$, but can't do much else with it.
Thanks for ...
1
vote
1answer
34 views
Characteristic polynomial of the differentiation map
Determine the characteristic and minimum polynomial of the differentiation map $D: \mathbb{R_n}[X]\longrightarrow \mathbb{R_n}[X]$ (where $\mathbb{R_n}[X]$ is a set of polynomials of degree at most ...
2
votes
3answers
21 views
How to form a cubic equation with the substitution method?
I had this question:
"Find the cubic equation whose roots are twice the roots of the equation $3x^3 - 2x^2 + 1 = 0$"
In my first attempt, I solved it through the use of simultaneous equations, where ...
1
vote
1answer
38 views
Arrangements of affine hyperplanes
Fix $n>0$ and $X\subseteq\mathbb{R}^n$. Call a function $f:X\longrightarrow \mathbb{R}$ linear if it is of the form
$$
f(\bar{x})=a_1x_1+\ldots+a_nx_n+b
$$
for some $a_i,b\in\mathbb{R}$.
Now ...
4
votes
1answer
36 views
Dimensions of vector subspaces in a direct sum are additive
$V = U_1\oplus U_2~\oplus~...~ \oplus~ U_n~(\dim V < ∞)$ $\implies \dim V = \dim U_1 + \dim U_2 + ... + \dim U_n.$ [Using the result if $B_i$ is a basis of $U_i$ then $\cup_{i=1}^n B_i$ is a basis ...
0
votes
1answer
49 views
Is it possible that $V=\cup_{i=1}^n V_i?$
Let the vector space $V$ be decomposable into the non-zero subspaces $V_i;~i=1(1)n.$ Is it possible that $V=\cup_{i=1}^n V_i?$
1
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0answers
37 views
Extending a rational entry matrix to an orthogonal matrix.
Suppose $M \leq N$ are positive integers and let $r_1,\ldots,r_M$ represent the rows of an $M\times N$ matrix $A$ in which: (i) the rows of $A$ are orthogonal and having the same norm, (ii) the ...
1
vote
2answers
25 views
Matrix involving values of polynomials
I've been doing this problem but im stuck.
Be $f_1 f_2 f_3 \in \mathbb{R}_2$[$x $]. Proove that {$f_1$,$ f_2$,$ f_3$} form a base of $\mathbb{R}_2$[$x $] as $\mathbb{R}$ vector space, if and only ...
1
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1answer
41 views
A question on linear operators
This is a problem I’ve been working on as part of my studies for an upcoming comprehensive exam:
Let $F$ be a field, let $V\in F$-$\mathrm{Mod}$ be a finite-dimensional left $F$-vector space, and let ...
3
votes
2answers
27 views
Linear Algebra determinant and rank relation
True or False?
If the determinant of a $4 \times 4$ matrix $A$ is $4$
then its rank must be $4$.
Is it false or true?
My guess is true, because the matrix $A$ is invertible.
But there is ...
1
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2answers
32 views
Linear Transformation Orthogonality
True or False:
If $T$ is a linear transformation from $R^n$ to $R^n$
such that
$$T\left(\vec{e_1}\right), T\left(\vec{e_2}\right), \ldots, T\left(\vec{e_n}\right) $$
are ...
0
votes
1answer
48 views
What are the two main ways to prove that a matrix $\Bbb R^{n\times n}$ is definite positive.
What are the two main ways to prove that a real $n\times n$ matrix is definite positive?
Is the first way: If a matrix is $n \times n$ symmetric matrix, then the associated quadratic form is ...
3
votes
1answer
48 views
Quadratic Equation with “0” coefficients
Let's say I have two objects $x$ and $y$ whose position at time $t$ is given by:
$$
x = a_xt^2+b_xt+c_x \\
y= a_yt^2+b_yt+c_y
$$
And I want to find which (if any) values of $t$ cause $x$ to equal ...
1
vote
1answer
20 views
Finite difference method stability
I have shown that a finite difference method satisfies
$$\underline{u}^{n+1}=((1+6\mu)\mathbb{I}-36\mu A^{-1})\underline{u}^n$$
I don't think that the rest of the question is necessary but it is ...
1
vote
1answer
27 views
The relationship between plane curves and the derivative of the Wronskian
I have found a theorem but I did not understand the proof. I'm looking for a clarification of the proof or a different proof.
Let $f_1, f_2, f_3$ be the three components of a curve in $R^3$ ...
4
votes
4answers
61 views
if $\mathbf x$ is sampled randomly from a hypercube on $R^n$, what is the probability density for $|\mathbf x| = d$
if the vector $\mathbf x$ is sampled randomly from a uniform distribution on $[0, 1]^d$, what is the probability density function for $|\mathbf x|$? Is it easy to scale for $[0, n]^d$?
2
votes
2answers
25 views
Showing set of tensored states span a space
I have the four states
$$
\lvert1\rangle \lvert1\rangle - \lvert0\rangle \lvert0\rangle \\
i\lvert1\rangle \lvert1\rangle + i\lvert0\rangle\lvert0\rangle \\
\lvert0\rangle \lvert1\rangle + ...
2
votes
2answers
47 views
Smooth maps on a manifold lie group
$$
\operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\
\begin{align}
&n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\
&n = 2, \operatorname{GL}_n(\mathbb ...
1
vote
0answers
12 views
Conditions for matrix operator to preserve complex symmetry on DFT vector?
Suppose there is a DFT vector $\mathbf{X}$ (complex vector) with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. ...
1
vote
1answer
30 views
Why don't we consider the zero subspace (which is readily $T$-invariant) in the definition of direct sum of linear operators?
Why don't we consider the zero subspace (which is readily $T$-invariant) in the definition of direct sum of linear operators?
REF: Schaum's Outline of Linear Algebra
4
votes
1answer
33 views
Book on quadric surfaces with linear algebra
Most information that I can find about quadric surfaces is written from a calculus perspective - without using any matrices or vectors. However, I would like to have a reference that tells me the ...
1
vote
0answers
30 views
Eigenbasis of a Hilbert space: isomorphism
Let $K$ be a matrix containing the dot product between points in a Hilbert space $\mathcal{H}$ (assume that it is finite-dimensional). Then, we could form a basis using the eigenvectors of a normal ...
1
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2answers
34 views
Solving the domain and range of a region satisfying two inequalities?
The question I was provided was:
"Find the domain and range of the region satisfied by the following inequalities:
i) $y \ge (x-1)^2$
ii)$y \le2x+1$
Any help would be greatly appreciated. Would you ...
0
votes
0answers
32 views
please help me with this question [duplicate]
How can I solve that? Thanks!
Let V = span{1, x, x2 , x3 } be a real inner product space with the
inner product defined by (f, g) = integral from -1 to 1 (fg)dx. Check that φ(f ) = f (0) is a
linear ...
3
votes
2answers
72 views
inner product space and polynomial
Let $V = \mathrm{span}\{1,x,x^2,x^3\}$ be a real inner product space with the
inner product defined by
$$
\langle f,g\rangle =\int\limits_{-1}^{1} fg
$$
Check that $T(f) = f(0)$ is a linear ...
0
votes
1answer
24 views
Column entries of a matrix sum to zero, so what are the properties?
What kind of properties does a matrix whose column entries sum to zero have?
$$ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & ...
2
votes
0answers
29 views
Resolving mass of holy disk with moment of inertia?
A uniform lamina of mass m is bounded by concentric circles with centre O and radii a and 2a. the lamina is free to rotate about a fixed smooth horizontal axis T which is tangential to the outer ...
4
votes
1answer
99 views
Linear equations; real solution; rational solution?
I saw this question
Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $B ∈ \mathbb{Q}^m$. Suppose
that the system of linear equations $AX = B$ has a solution in
$\mathbb{R}^n$. Does it necessarily have ...
