Tagged Questions
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 ...
0
votes
2answers
12 views
Finding a rotation matrix
I am looking for a rotation matrix such that
$$
\operatorname{rot} \cdot
\begin{pmatrix}
-1 & 0 & 0 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1
\end{pmatrix}
=
...
1
vote
0answers
8 views
Does the Duality Theorem of Linear Programming hold only in closed convex cones
I've just read the the Duality Theorem of Linear Programming. Here is the proof from my book (and my questions after it):
Duality Theorem of Linear Programming: If the primal or dual linear ...
1
vote
1answer
19 views
Optimize matrix arrangement
Let's imagine I have a Matrix $\textbf{C}$ whose construction depends on several parameters (and constraints).
I'm interested in maximizing a value $K$ calculated as:
$K=\frac{-1}{C_{1,1}^{-1}}$
...
2
votes
0answers
24 views
How prove this stronger Cauchy-Schwarz inequality for traces of compression matrices
Question:
Assume that $A$ and $B$ are contractions, so
$I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let
$C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that:
...
1
vote
2answers
32 views
Some questions about the proof of the General Linear Group being a manifold.
I understand the idea behind proving that GL(n,$\mathbb{R}$) is a smooth manifold by first using the fact that it is isomorphic to $\mathbb{R}^{n^{2}}$ and using the continuity of the determinant ...
0
votes
1answer
35 views
Allow $2 \Bbb N$ to denote the even integers $> 0$.
Please help!
Allow $2\Bbb N$ to denote the even integers $> 0$. Say $a \in 2\Bbb N$ is irreducible if there are no numbers $b, c \in 2\Bbb N$ so that $a = bc$.
(1) Show that if $n$ is an odd ...
-1
votes
1answer
22 views
Solve 6 = 13 - IE - ((13 - IE) * .4)
I'm embarrassed to ask this question. How do I solve this equation?
6 = 13 - X - ((13 - X) * .4)
My next step would be...
...
5
votes
3answers
337 views
Meaning of math symbol ~
Segment of Example:
t = ...
More usefully, we have:
t ~ n*log(n)
Note: ~ means "similarity" like in geometry, same shape but not same size. How is it interpreted here?
Edit: yes, t depends on n
...
1
vote
1answer
11 views
Issue understanding the difference between reduced row echelon form on a coefficient matrix and on an augmented matrix
I have a problem understanding something about matrices and the difference between a coefficient matrix and an augmented matrix. One theorem in a book I'm reading states:
Suppose thate $A\mathbf{x ...
2
votes
0answers
46 views
Prove that the norm of $E$ is generate by the inner product $\langle x,y \rangle =\frac{1}{4}\left(||x+y|^2-||x-y||^2\right)$
Let $E$ a normed linear space such that:
$$\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2 $$
Prove that the norm of $E$ is generate by the inner product
$$\langle x,y \rangle ...
0
votes
2answers
31 views
Calculate the dimension of $U = \{(x_1,x_2,x_3,x_4,x_5) : x_1+x_3+x_5=x_2+x_4=0\}$
In the vector space $V \subset \Bbb R^5$, considering the vectors $v_1,v_2,v_3$
$v_1 = (0,1,1,0,0)$
$v_2 = (1,1,0,0,1)$
$v_3 = (1,0,1,0,1)$
We have $V = \mathrm{span}(v_1,v_2,v_3)$ ...
1
vote
2answers
33 views
Does it span $\mathbb{R}^3$?
I have a T/F question and I think I know the direction to go, but I am not sure.
It states: $\{[17,6,-4]^t,[2,3,3]^t,[19,9,-1]^t\}$ does not span $\mathbb{R}^3$.
Let me get this straight. It SPANS ...
0
votes
0answers
21 views
Linear independence of linear maps [on hold]
Consider the $R$-vector space of the maps from $[0,4]\rightarrow R$, and the subspace $W\langle f_1,f_2,f_3,f_4\rangle$ of F, such that:
$f_1(x)=\begin{cases}1 \mbox{ if } x\in[0,2]\\0 \mbox{ if } ...
4
votes
3answers
85 views
Is Matrix $A^2$ invertible if $A$ is invertible?
I want to say that squaring is a form of scaling so that it should be true; however I can't make sense out of it and clearly see why.
The problem is that while it's a form of scaling, if we think ...
1
vote
1answer
42 views
About $\mathbb{F}_7[x]$
can you help me with this?
Let $a(x)=3x^6+2x^2+x+5$ and $b(x)=6x^4+x^3+2x+4$, find the g.c.d between $a(x)$ and $b(x)$ in $\mathbb{F}_7[x]$.
Thanks!
0
votes
2answers
31 views
Show that $A+BC^T$ is nonsingular, and that$ (A+BC^T)^{−1} = A^{−1}−A^{−1}B(I +C^TA^{−1}B)^{−1}C^TA^{−1}$
Ok, so I've been trying to work on this problem:
Let $n$ and $p$ be positive integers and let $\Bbb F$ be a field.
Let $A$ \in $M_{n\times n}(\Bbb F)$ and let $B,C \in M_{n\times p}(\Bbb F)$ be ...
-1
votes
0answers
13 views
basis vectors of a 2D lattice plane in a 3D lattice
I know the basis vectors of the three-dimensional lattice $\Lambda = \{\mathbf{b_1}, \mathbf{b_2}, \mathbf{b_3} \}$. I also know the equation of the plane in this 3D lattice, suppose $Ax + By + Cz = ...
2
votes
3answers
75 views
Center of $GL_n(\mathbb R)$ is the set of matrices $\lambda I$
I determined the set of all matrices $A$ such that $AB = BA$ for all $B$ in $GL_n(\mathbb R)$ to be the set of $\lambda I$. Now I'm not sure this is true. But quite sure. So I tried to prove it and it ...
-1
votes
2answers
45 views
Similar matrices that are not diagonalizable
Let $f:R^3 \to R^3$ be a function which matrix with respect to the standard basis is:
$$
A = \begin{pmatrix}
-4 & -8 & 8 \\
1 & 2 & -2 \\
-1 & -2 & ...
0
votes
0answers
10 views
Homogeneous system of equations , and sub-set K of $R^4$
Given K,L are sub-sets of $K^4$:
$K = \{(-5,8,14,0),(-1,4,2,4)\}, L = \{(0,1,-10,8),(0,3,-1,5)\}$
Find a homogeneous system of equations that its solutions are Spanned by K.
Also prove that ...
3
votes
3answers
129 views
If $A=AA^{\top}$, show that $A^2=A$
I've been working trying to understand the following question:
Let n be a positive integer, let $F$ be a field, and let $A \in \mathrm{Mat}(n,F)$ satisfy the condition $A=AA^{\top}$. Show that ...
4
votes
0answers
41 views
mortality problem
The mortality problem is the question if some product of a given set of matrices
yields the 0-matrix. In general, the mortality problem is undecidable. To have a
feeling for the difficulty of the ...
0
votes
1answer
62 views
Compute the exponential of a matrix
How we can show that
$$e^{At}=\frac{e^{-\lambda t}}{\mu}[(\mu\cos\mu t+\lambda\sin\mu t)I +A\sin(\mu t)]$$
where
$$\lambda=\frac{a_0}{2},\qquad
...
3
votes
3answers
87 views
If $AB = I$ then $BA = I$: is my proof right?
I want to prove that for matrices $A,B \in M_n (\mathbb K)$ where $\mathbb K \in \{\mathbb R, \mathbb C, \mathbb H\}$ if $AB = I$ then $BA = I$.
My proof is really short so I'm not sure it's right:
...
3
votes
0answers
83 views
$n$ distinct real eigenvalues of an $n \times n$ matrix
What are the necessary and sufficient conditions for a real $n \times n$ matrix to have $n$ distinct real eigenvalues?
Ideally I'm looking for a test that does not require (and is hopefully more ...
1
vote
0answers
21 views
Given matrix equation $A x = b$, find new matrix $A^*$ after permutation switching $x_i \leftrightarrow b_i$
Given matrix equation $A x = b$, where $A$ is square, symmetric matrix, how to find new matrix $A^*$ such that:
$A^* x^* = b^*$
and $x^*$, $b^*$ are created by switching vector elements $x_i ...
2
votes
1answer
18 views
Geometric characterization of an Euclidean norm
Show that $N$ is an Euclidean norm if and only if the intersection of the unit ball with any plane is an ellipse.
I'm stuck on this one. I do not see how can I connect the definition of an ...
2
votes
2answers
90 views
Determinant involving recurrence
Evaluate
$$\left| A \right| = \left| {\matrix{
{x + y} & {xy} & 0 & \cdots & \cdots & 0 \cr
1 & {x + y} & {xy} & \cdots & \cdots & 0 \cr
0 ...
2
votes
2answers
56 views
Prove $\dim W \ge 2$
Let $U_1, U_2, W$ subspaces of a finite dimensional vector space, such that:
$U_1 \cap U_2 = \{0\}$
$U_1 \cap W \ne \{0\}$
$U_2 \cap W \ne \{0\}$
Show that $\dim W \ge 2$.
...
0
votes
2answers
32 views
How to see a matrix presents a linear transformation?
Consider the transformation $T:P_n \rightarrow P_n$ ($P_n$ is the vector space of polynomials of degree at most $n$, with complex coefficients) and its associated matrix presentation, namely $F$. How ...
0
votes
1answer
25 views
If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable.
If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable.
It seems very obvious intuitive, but how to write a good solid proof? Notice I take the closure of the span (the ...
1
vote
1answer
31 views
Subspaces of the set of real valued functions over an interval.
Show that the integral of all continuous real-valued functions on the interval [0,1] equal to b $\in$ R is a subspace of $R^{[0, 1]}$ if and only if b=0.
So I am assuming that because both the ...
0
votes
1answer
16 views
whats the general step by step formula for finding an equation for a plane passing 2 points perpendicular to a plane?
i've checked multiple places for a general formula to follow including here:
Find an equation of the plane passing through 2 points and perpendicular to another plane
I know it asks the same ...
1
vote
1answer
36 views
Finding the closest point in a set to another point in n-dimensional space: efficiently
I'm a programmer and am working on writing an efficient algorithm that, given a point P in n-dimensional space, can find the closest point from a set of points. For ...
0
votes
3answers
50 views
Complex Numbers and Linear Algebra
Explain why there does not exist a $\lambda $ in the Complex Field such that
$$\lambda \left(2-3i, 5+4i, -6+7i \right) = \left(12-5i, 7+22i, -32-9i \right)$$
Can someone help me figure out how to go ...
0
votes
2answers
30 views
Finding multiple solution of a matrix
I have this matrix from a book's exercise.
$$
\left[
\begin{array}{@{}cccc@{}}
a&0&b & 2 \\
a& a& 4 & 4 \\
0&a& 2 & b\\
\end{array}
\right]
$$
be the augmented ...
11
votes
3answers
419 views
Expected Value of a Determinant
Suppose that I construct an $n \times n$ matrix $A$ such that each entry of $A$ is a random integer in the range $[1, \, n]$. I'd like to calculate the expected value of $\det(A)$.
My conjecture is ...
2
votes
1answer
55 views
Determining whether or not a vector is a linear combination of a give matrix
$$
A=
\begin{bmatrix}
1 & 0 & 5\\
-2 & 1 & -6\\
0 & 2 & 8
\end{bmatrix}
,b= \begin{bmatrix}
2\\
-1\\
6
\end{bmatrix}
$$
The problem asks to determine whether or not vector $b$ ...
5
votes
0answers
46 views
Reference Request: Prereqs for Lecture Notes on “Abstract Linear Algebra”
I just found this set of lecture notes on linear algebra which seems to go over several things I've been wondering about as I study linear algebra. Unfortunately there are very few exercises in the ...
1
vote
1answer
10 views
Linear Independence of vectors in relation to the $t$ parameter
Considering the following vectors $u,v,w\in R^5$
$u = (0,1,1,-1,1/2)$ $v = (t,1,1,0,1)$ $w = (1,3,3,-2,-2)$
Then, they are linearly independent:
a) for each value of $t$ b) if and ...
2
votes
2answers
65 views
Dot product notation
Let $\mathbf{A=(a_1,a_2,\ldots, a_n)}$ and $\mathbf{B=(b_1,b_2,\ldots,b_n)}$. Many linear algebra books and texts define the dot product as
$$
\mathbf{A\cdot B^T=a_1b_1+a_2b_2+\cdots+a_nb_n}
$$
where ...
0
votes
1answer
41 views
About a vector space $F^\infty$
$F^\infty$ is a vector space defined as $\{(x_1, x_2...) : x_j \in F$ for $j = 1, 2, \ldots, n\}$. Is it correct?
A list can't be infinite in length, but an element of $F^\infty$ looks to be an $ ...
3
votes
2answers
36 views
Show $rk(A) + rk(B) \ge rk(A+B)$
Show $rk(A) + rk(B) \ge rk(A+B)$, where $A,B \in M_{m\times n}(\mathbb{F})$
I'm trying to think in terms of linear transformations.
We can define $T_a, T_b:\mathbb{R}^n\rightarrow \mathbb{R}^m$
I ...
0
votes
0answers
20 views
Finding equations for subspace spanned by given vectors
I have no idea how to deal with special operations within a subspace. How should I approach the problem? Can I use Gauss and Row echelon form to see if the vectors are LI or LD? (Will that even help ...
1
vote
1answer
34 views
Signed determinant of quadratic forms over Q_p
Let $W(k)$ be the Witt-Ring of the field $k$.
in this script http://math.uga.edu/~pete/quadraticforms2.pdf at the bottom of page 2 the signed determinant is introduced by
$d^\pm (q) = ...
4
votes
1answer
26 views
$rk(A)=n$ implies $rk(AB)=rk(B)$
Let $A \in Mat_{m\times n}(\mathbb{R})$ and $B \in Mat_{n\times p}(\mathbb{R})$.
Assume $rk(A)=n$. Prove that $rk(AB)=rk(B)$.
Lets start by proving $rk(B) \ge rk(AB)$. Indeed, since the ...
0
votes
3answers
46 views
Show That the Linear Transformation is NOT linear
So I'm presently reviewing my teachers optional notes and I cannot seem to understand how I can show that the following transformation is NOT linear. I don't need a complex proof, just a short ...
1
vote
3answers
27 views
Find that the given linear transform is a isomorphism
I'm studying Linear Algebra and I'm having trouble demonstrating that a function is a isomorphism, that is:
"Given the linear transform $T: V \rightarrow W$, $T$ is a isomorphism if and only if it is ...
2
votes
2answers
51 views
Show $L_1 \subseteq L_2$ or $L_2 \subseteq L_1$
Let $L_1,L_2$, two subspaces of a finite dimensional vector space.
Prove that if $\dim(L_1 + L_2) = 1 + \dim (L_1 \cap L_2)$ then $L_1\subseteq L_2$ or $L_2 \subseteq L_1$.
Well, I've read a ...
1
vote
0answers
19 views
Multiplication of series [on hold]
Let $S_\infty$ and $T_\infty$ be two infinite series.
Let $ S = \sum _{r = 1}^{\infty} x_r $
Under what conditions could you say that :
$$S_\infty \times T_\infty = \sum_{r=1}^{\infty} (x_r \times ...
