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)

0
votes
1answer
25 views

Theorem regarding direct sums

Let $w_1$ and $w_2$ be subspaces of V. Prove that V is direct sum of $w_1$ an $w_2$ iff each vector in V can be uniquely written as $x_1 + x_2$ where $x_1$ belongs to $w_1$ and $x_2$ belongs to $w_2$ ...
0
votes
1answer
21 views

Construction of a Module isomorphism

Let $R$ be a PID. Consider the sets $X_0=\{v_0,v_1,v_2\}$ and $X_1=\{e_1,e_2,e_3\}$ and let $C_i$ be the free $R$-module on $X_i$ for $i=0,1$. Consider the $R$-module homomorphism $$C_1\;\...
1
vote
1answer
43 views

Overlap between two vectors

Given are two vectors ${\bf g}_1, {\bf g}_2\in\mathbb{R}^N$ with non-zero scalar-product ${\bf g}_1^\top{\bf g}_2 \ne 0$. Then there exist three unique (up to the sign) orthonormal unit vectors ${\bf ...
1
vote
0answers
33 views

How to randomly generate two integer matrices $A$ and $B$, so that entries of 3 metrics $A$, $B$, and $AB$ are within certain range?

I ran into this question when writing a program. I need to generate two matrices, and calculate their product. However, I must ensure all entries are within 8-bit signed integer range, i.e. $[-128, ...
0
votes
0answers
20 views

Water drop evaporation time and contact angle

I'm measuring water drop evaporation on different surfaces and it would be nice to have an equation to roughly estimate evaporation time (or contact angle). Some drops are hydrophobic, others ...
1
vote
0answers
39 views

What is a linear isomorphism?

I am working with the book Manifolds and Differential Geometry from Lee and I am a little bit puzzled since he sometimes talks about linear isomorphism (proposition 2.3 for example). But isn't an ...
1
vote
0answers
40 views

Finding a linear system to solve quadratic equations

considering an equality with a polynomial of second degree where the coefficient for $x^2$ is $1$ I know that $$ a x^2 + b x + c = a(x-\alpha)(x-\beta) = 0 $$ I also know that $$ \alpha + \beta = -...
1
vote
2answers
80 views

How come two of the eigenvalues are same?

Question is about finding the eigenvalues of the matrix : $$\begin{bmatrix} 0 & 0 & 2 \\ 0 & 2 & 0 \\ 2 & 0 & 0 \\ \end{bmatrix}$$ the matrix would become $$\begin{bmatrix} -...
2
votes
1answer
36 views

Least Squares Algorithm with Inverse Norm

Given an overdetermined linear system $A \in \mathbb{R}^{m \times n}$, $b \in \mathbb{R}^{m \times 1}$ with $A < 0$ and $b < 0$. What is a good way to numerically determine $$ \min_x \left\lVert ...
1
vote
0answers
38 views

Real and imaginary part of tensors of matrices

Given a matrix $A\in \mathbb{C}^{n\times m}$, clearly we can write $A=\Re(A)+i \Im(A)$, i.e., the real and imaginary part of $A$. (For instance, $A=[1,i]$, then $A=[1,0]+i[0,1]$). I am interested in ...
-2
votes
0answers
40 views

what is Expected Mean

Thus the expected mean $\mu$ of the set $\mathcal S$ can be given as \begin{align*} \mathbb E \mu&= \sigma^2+\frac 1r \sum_{i=1}^m\left(\mathbb E\lambda_i-\sigma^2\right)\\ &\geq \sigma^2+\...
0
votes
1answer
19 views

Calculating the missing two points of rectangle if 2 points and the aspect ratio are known

How can I calculate the missing two points of a rectangle if I know 2 points (top left and top right) and the aspect ratio i.e 16:10. For example: Top left: A(834, 449) and Top right: B(1675, 423)
2
votes
1answer
16 views

Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form?

Task: Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form? Solution: Since a Minkowski-form has the type $(n - 1, 1)$, ...
0
votes
0answers
33 views
0
votes
1answer
30 views

Question on proof of number of solutions of linear system

The proof my book uses starts off by saying: "If the system has exactly one solution or no solutions, then there is nothing to prove", and then continues on by assuming there is an infinite ...
1
vote
1answer
41 views

prove subspace of a vector space

Consider the subset $T$ of $\mathbb{R}^2$ defined as follows: $T := \left\{(x, y) : x, y \in \mathbb{R} : y = 3x \right\}$. Prove that T is a subspace of the vector space $\mathbb{R}^2$. My attempt: ...
3
votes
3answers
78 views

Which non-negative matrices have negative eigenvalues?

It's easy to proof by counterexample that non-negative matrices can have negative eigenvalues. For example, the following matrix have -1 as an eigenvalue: $$ A = \begin{bmatrix} 0 & 0 & 0 ...
1
vote
0answers
43 views

Polar coordinate in Cartesian

My book states one can write polar coordinates $(\hat{r}, \hat{\theta})$as $$\hat{r} = \cos \theta i + \sin \theta j$$ $$\hat{\theta} = -\sin \theta i + \cos\theta j$$ Can someone explain how $$\hat{...
0
votes
1answer
21 views

Is there a unique projection map in this case?

Let $X$ be a Banach space over $\mathbb{C}$. Let $A,B$ be closed subspaces of $X$ such that $X=A\oplus B$. Assume that $||a+b||=||a||+||b||$ for each $(a,b)\in A\times B$. Then, does there exist a ...
1
vote
2answers
47 views

Finding an orthonormal basis for the plane $x_1 - 5x_2 - x_3 = 0$

Find an orthonormal basis of the plane $x_1 - 5x_2 - x_3 = 0$ I'm having trouble with this problem. So I picked the vectors $u_1 = \begin{bmatrix}1\\0\\1\end{bmatrix}$ and $u_2 = \begin{bmatrix}5\\...
1
vote
1answer
47 views

How to project $x_2$ onto $u_1$

I'm following a solution from here (the first problem), I don't understand how to "project $x_2$ onto $u_1$" 1) how does:$\begin{bmatrix}0\\\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\end{bmatrix}$ ...
0
votes
1answer
20 views

Choosing the seed for a LFSR

I was just wondering how the seed of a LFSR is chosen and is there any connection between the seed chosen and cryptographic strength of the keystream generated? Thank you
3
votes
3answers
59 views

If $T$ and $T^2$ have equal rank then $V=\ker T\oplus {\rm im}\, T$ for $V$ finite dimensional.

I am trying to prove the following: Let $V$ be a finite-dimensional vector space. Consider an operator $T$ on $V$ such that $\text{dim range}(T)=\text{dim range}(T^2)$. Show that $V=\text{null}(T)\...
0
votes
1answer
29 views

How to rotate a coordinate system in $\mathbb{R}^3$ through an angle about an arbitrary axis passing through origin?

The question spurred in my mind when I was asked the following: Find the transformation matrix T that describes a rotation by $120^\circ$ about an axis from the origin through the point $(1,1,1)$....
-2
votes
1answer
35 views

Does the line pass through the origin? [on hold]

Does the line $L: (3,4,-1)+s(7,-2,1),\ s\in \Bbb R$ pass through the origin? I'm not sure how to do this.
-4
votes
0answers
26 views

Conjugacy-classes in $GL_n(\mathbb{Z}/p\mathbb{Z}) × GL_m(\mathbb{Z}/q\mathbb{Z})$ [on hold]

Find the number of conjugacy-classes in $GL_n(\mathbb{Z}/p\mathbb{Z})× GL_m(\mathbb{Z}/q\mathbb{Z})$ of cyclic subgroups of order pq?.
0
votes
2answers
36 views

Determine the distance between the point $(2,-3,1)$ and the point of intersection of three planes

Determine the distance between the point $(2,-3,1)$ and the point of intersection of the following system. $3x-y+z=4$ $-x+2y+3z=7$ $x+3y+4z=12$ I'm not quite sure how to do this?
3
votes
2answers
23 views

(Double) Coset of $GL(n, q^2)/GL(n, q)$

I am trying to understand a particular coset/double coset of the finite group $G = GL(n, q^2) = GL_n(\mathbb{F}_{q^2})$. It has a natural subgroup $H = GL(n, q)$, which can also be viewed in the ...
2
votes
0answers
30 views

Geometry Of Unitary Transformations

Ever since I first took Linear Algebra, I have over time realized how concepts like determinants, eigenvalues, diagonalization, orthogonal transformations and so on have very intuitive geometric ...
0
votes
1answer
22 views

Linear maps uniqueness proof: Difference between “uniquely determined on span($v_1,…,v_n$)” and “uniquely determined on V”

I've just been introduced to linear maps, and I'm still trying to wrap my head around the proofs, which don't read easily for me. In the uniqueness proof, the very last section states: Thus $T$ (the ...
3
votes
2answers
76 views

Why doesn't transposing matter?

I was helping a friend with linear algebra, particularly I was teaching how to check if a collection of vectors $v_1, ..., v_k \in \mathbb{R}^n$ are linearly independent (assuming the vectors are ...
4
votes
2answers
52 views

Intuitive understanding of the matrix of a linear transformation

Is it accurate to say that a matrix $M(T)$ of the linear map $T:V\to W$ encodes the linear map into a series of numbers by showing how the linear map applied to the basis vectors of $V$ can be ...
1
vote
2answers
26 views

Number of multisets with restrictions on specific element count

I am looking to find the number of multisets with restrictions on the number of specific elements. This isn't for homework, it is a work related problem. My set of items is {A, a, B, b}. I want to ...
0
votes
0answers
13 views
2
votes
0answers
31 views

Linear Algebra Over Integer (Book Recommendation)

I am doing research on Integer Spline. So, there are lots of system of linear diophantine equations appear, which I have to solve them within the integer. I am lacking the theory behind the system of ...
2
votes
0answers
36 views

Relationship between eigenvalues of Hermetian matrices

Suppose that we have two $m\times m$ matrices $A$ and $B$ which are Hermetian, with $|B_{ij}|\leq |A_{ij}|$ for $i,j = 1,2 \cdots, m$. Can we say anything about the relationship between the largest ...
2
votes
0answers
44 views

Is there always $n$ permutations of a vector in $R^n$ that are linearly independent?

As long as the $n$ entries of the vector are all different and they dont add up to zero. If it is true, how to prove it, if not, what is a counter example?
0
votes
1answer
14 views

Solving the Price of Widgits

So I've got a word problem. A man is selling widgits on the street corner. The price for one widgit is $50. However, for every widgit he sells, he increases the price by 5%. What is the total amount ...
1
vote
2answers
20 views

A Vector of acute angle to all vectors in a set

I´m interested in the following problem: Given a set $A$ of vectors in $R^n$, find out whether it is possible to find a vector $v$ s.t. $$\forall a \in A: v\cdot a \gt 0,$$ or in other words, the ...
-1
votes
2answers
33 views

linear algebra - raising a matrix to a certain degree

If you are given any matrix A with certain entries and are asked to compute the same matrix raised to any power such as k, are you supposed to raise each entry in the matrix to the k-th degree?
0
votes
2answers
74 views

Is true that if $V \cong W$ and $Z \cong Y$ then $V\times Z \cong W \times Y?$

Let $V,W,Z,Y$ finite dimensional vector spaces. Is true that if $V \cong W$ and $Z \cong Y$ then $V\times Z \cong W \times Y?$ My attempt: Suppose it is true: Then there is $f : V \to W$ isomorphism ...
2
votes
2answers
46 views

How to show using the universal property that $V\otimes V^{\ast} \cong \mathcal{L}(V,V)$?

Let $V$ a vector space of finite dimension and $V^{\ast}$ its dual space. How to use the universal property to show that $V\otimes V^{\ast} \cong \mathcal{L}(V,V)?$ I just know that I can construct ...
1
vote
1answer
31 views

Basic questions about optimizing concave function with constraints

Consider the following problem: \begin{align} {\tt Maximize} \quad M(\mathbf y)& = \log \Big(\prod_i U_i(y_i) \Big) \\ y_i & = \sum_{j=1}^{m} \frac{x_{ij}}{a_{ij}} \\ \sum_{i=1}^{n} \frac{c_i ...
8
votes
1answer
89 views

Prove that matrix $A$ diagonalizable if $A^2=I$ using characteristic polynomial

Prove that the matrix $A$ is diagonalizable if $A^2=I$ using characteristic polynomial I saw an answer that used the minimal polynomial of $A$. Can that be proven without using minimal polynomial? ...
1
vote
1answer
20 views

Example of 2 trinomial multiplication which is equal to sum of 2 monomials

How to find out $P$ as an algebraic monomial which $P=ma$ and $(a^2 P+1)(a^2 P+1)$ answer be sum of two monomials $Q,R$ eg $(a^2+a+1)(a^2-a+1) = a^4 + a^2 + 1$ which is sum of three monomials. *...
2
votes
3answers
48 views

A set of linear algebra questions?

Could you help me with these questions, I figured most of them out on my own, but I'm not completely sure if I'm correct. a) $A=\begin{bmatrix}a^2&ab&ac\\ ab&b^2&bc\\ ac&bc&c^...
3
votes
3answers
91 views

How to find the determinant of this $n \times n$ matrix in a clever way?

\begin{bmatrix} b_1 & b_2 & b_3 & \cdots & b_{n-1} & 0 \\ a_1 & 0 & 0 & \cdots & 0 & b_1 \\ 0 & a_2 & 0 & \cdots & 0 &...
1
vote
5answers
151 views

Good true-false linear algebra questions?

Can you suggest me a collection of true-false linear algebra questions, like the ones found in the MIT exams, if possible with solutions (i.e. explanations)? Sorry if it turns out that my request is ...
6
votes
2answers
279 views

Solution(s) to dot product of vectors

I have some questions about the uniqueness of matrices when post- and pre-multiplied with vectors (inner product). Say we have two vectors $\vec{a}$ and $\vec{b}$, whose inner product is a scalar, ...
6
votes
1answer
45 views

Do eigenvalues of a linear transformation over an infinite dimensional vector space appear in conjugate pairs?

While attempting to answer a question here (namely, the finite dimensional case of the title question: Prove that if $\lambda$ is an eigenvalue of $T$, a linear transformation whose matrix ...