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 ...

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3
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2answers
117 views

Prove that the union of three subspaces of $V$ is a subspace iff one of the subspaces contains the other two.

Prove that the union of three subspaces of V is a subspace iff one of the subspaces contains the other two. I can do this problem when I am working in only two subspaces of $V$ but I don't know how ...
-3
votes
2answers
32 views

If $F_3 =\mathbb{ Z}/3\mathbb{Z}$, show that $F_3$ is a field. How can this be done? [closed]

If $F_3 = \mathbb{Z}/3\mathbb{Z}$, show that $F_3$ is a field. How can this be done? Please help! $\mathbb{Z}=\{ \text{set of integers}\}$.
0
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2answers
44 views

Show that if K is a non-zero ideal of Z/mZ,

Show that if K is a non-zero ideal of Z/mZ, then K is the principal idea. Please help!
0
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3answers
23 views

In each part, find a basis for the given subspace ofR 3 , and state its dimension

guys I gotta be honest, I've taken notes on everything in the last two sections for this but I'm not sure how to find a basis for a subspace that is a lone plane/line etc.. a full explanation would ...
1
vote
0answers
21 views

In Exercises1–6, find a basis for the solution space of the ho mogeneous linear system, and find the dimension of that space.

for number 1, I tried reducing the matrix to solve for x1, x2, x3, but i got a row of 0's, so I'm not sure what to do after that... and what exactly if i have a solution? i saw an example of having ...
0
votes
1answer
52 views

Seeking Help on Linear Algebra Problem, Thank you all (YES or NO ANSWER)

Did i do the 1st one correct? Having difficulty understanding 2...1 step at a time just want to know if i did 1 correct
1
vote
1answer
39 views

Invariants of matrices

I am confused as to what this question is actually asking. I understand that if A and B are similar $P^{-1}AP=B$
3
votes
2answers
39 views

Surjective linear transformations: is checking for linear independency enough?

Is the linear transformation $T: \mathbb{R}^3\to\mathbb{R}^3$ defined as $$\begin{bmatrix} 0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & -1 \end{bmatrix}\begin{bmatrix} x \\ y ...
1
vote
1answer
22 views

$A$ is hermitian if and only if $\langle A\alpha,\beta\rangle= \langle\alpha ,A\beta\rangle$ for $\alpha$ and $\beta \in \mathbb{C}^n$

How can i prove that $A$ is hermitian if and only if $\langle A\alpha,\beta\rangle= \langle\alpha ,A\beta\rangle$ for $\alpha$ and $\beta \in \mathbb{C}^n$ i stuck in this problem i know that if $A$ ...
0
votes
0answers
25 views

Properties of real matrix in Schur Form

This question is from an old exam qualifier. 1.) Show that any $n \times n$ real matrix $A$, may be written as $A = QRQ^{*}$, where $Q$ is unitary and $R$ is upper triangular. Neither $Q$ nor $R$ ...
0
votes
2answers
28 views

Symmetric Matrix Quadratic Form

Let $A,B\in\mathbb{M}_{n\times n}(\mathbb{R})$ and $A,B$ are symmetric matrics. Prove that if $\vec{x}^TA\vec{x} = \vec{x}^TB\vec{x}$ $\forall\vec{x}$, then $A=B$. Since $A,B$ are symmetric, they are ...
1
vote
1answer
23 views

Prove or disprove a statement about testing the convexity of a set using the vertices

Assume we are working in $\mathbb R^d$. Let $A=\text{Conv}(V)$, the convex hull of $V$. Also $B=\text{Conv}(W)$. I am in a situation where I can prove the following: the line segment joining $v$ and ...
0
votes
0answers
28 views

Finding the largest eigenvalue of a sparse matrix

I would like to find the largest eigenvalue of a sparse matrix by hand- this is part of analyzing a mathematical model for infectious diseases. The nonzero entries are very complicated - hence Maple ...
0
votes
1answer
22 views

orthogonal matrix

I have to show the following claim: Let $A\in Mat(n,\mathbb{R})$ be positive definite and symmetric. Show that there exists a Matrix $T\in Mat(n,\mathbb{R})$ such that $T^tAT$ is a diagonal matrix. My ...
11
votes
4answers
332 views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...
7
votes
5answers
114 views

Positive semi-definite of a matrix composed of semi-definite blocks

Say a matrix A is positive semi-definite. Let B be a square matrix composed of replicas of A as sub-blocks, s.t. $$B=\begin{pmatrix} A & A \\ A & A \\ \end{pmatrix},$$ or $$\begin{pmatrix} A ...
1
vote
2answers
65 views

linear transformation of finite dimensional vector spaces

Let $V$ and $W$ be finite dimensional real vector spaces and $T\colon V\to W$ be linear. (a)Prove that if $\dim(V) < \dim(W)$, then $T$ cannot be onto. (b)Prove that if $\dim(V) > \dim(W)$, ...
6
votes
1answer
91 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
2
votes
1answer
61 views

Spectral norm proof (without the knowledge of eigenvector)

EDITED after found false Define $|v|$ as the normal Euclidean norm. $A$ is a n by n matrix. Define the spectral norm: $$|A| = \max \frac{|Ax|}{|x|}$$ I need to proove that: $|A|$ is the maximum of ...
0
votes
1answer
27 views

why is the coordinate vector of p = (7p1,-8p1,1p1) and not (7p1,-1p1,2p1)? (probably quick question)

why is the coordinate vector of p = (7p1,-8p1,1p1) and not (7p1,-1p1,2p1)? (probably quick question) I've looked at my book 5 times already to confirm the formula coordinate vector relevent to S ...
1
vote
0answers
16 views

Triangularisation of a linear map

The Calculation of the Char Poly is wrong but it's the method I am not able to understand In this example why does the eigenvector of A give the required eigenvector that is contained in the basis? ...
-3
votes
2answers
44 views

Help with vectors problem [closed]

Find unit vectors u1 and u2 in the directions of v=(3,1) and w=(2,1,2), find unit vectors u1 and u2 that the perpendicular to u1 and u2?
0
votes
4answers
57 views

How can I factor $x^2 + 2\sqrt{3}\,x + 3$? [closed]

$$x^2 + 2\sqrt{3}\,x + 3$$ Anyone could tell me how may I factor this? Thanks a lot
3
votes
2answers
28 views

Find value of $n$ with given conditions

The 4-digit positive number $n$'s digit sum is $20$. The sum of the first two digits is $11$, the sum of the first and the last digit as well. The first digit is the last digit $+3$. What is the ...
1
vote
1answer
26 views

Triangularisation of a linear transformation

I understand that Upper triangular matrices must have at least one eigenvector, but why does this mean that the basis of $[T]_B$ must contain an eigenvector for $[T]_B$ to be upper triangular?
7
votes
2answers
98 views

Proof of the inequality $\sqrt{\det X} \leq \frac{\operatorname{tr}X}{2}$

Let $A, B \in M_2(\mathbb{R})$ be symmetric and positive definite. Put $X:=AB$. then, we have the following inequality: $$\sqrt{\det X}\leq \dfrac{1}{2}\operatorname{trace}X.$$ and the equality ...
1
vote
0answers
33 views

Matrix multiplication in quaternions is not necessarily linear

I tried to show by example that matrix multiplication for quaternionic matrices is is not necessarily $\mathbb H$-linear. If $A \in M_n(\mathbb H)$ is a quaternionic matrix and $x$ is a vector in ...
0
votes
4answers
45 views

algebraic representation of a line in 3d

Is an algebraic representation of a line in 3d possible, or there can be only a parametric one?
0
votes
1answer
19 views

the volume of pyramid value

when calculating the volume of pyramid using a determinnat, is it ok to take the determinanat in absloute value so that every negative result would be converted to positive volume number?
1
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0answers
34 views

Schur decomposition of real-eigenvalue matrix

Is Schur decomposition of real-eigenvalue matrix a real orthogonal decomposition? If yes, why is it? Is it because all the eigenvectors are real? If I have $$ A^T+A^2=I $$ then I deduced ...
-1
votes
0answers
23 views

show matrices form a basis for vector space 2x2 (revisited) [duplicate]

matrices forms a basis for vector space 2x2 I have a question just like this to solve and the books example is generalized and is very vague about how to solve a specific problem. but in it, it said ...
1
vote
1answer
21 views

How to find distance between vector and a subspace

Well, this is question from a test that i had, i did'nt know how to answer it so i frorwarding this to you: Consider $v\:=\:\begin{pmatrix}\frac{1}{3} \\\frac{2}{3}\: \\\frac{2}{3}\end{pmatrix}$. ...
0
votes
1answer
18 views

Show that the following polynomials form a basis for P2 (confirm answer/method) (quick plz possibly?)

just wanna check if im doing htis right here: to check if its linearly independent i took x^2, x, x^0 to be K values, put their coeficients in a guasian problem and check if all k's are 0 like ...
1
vote
1answer
43 views

$A,B$ matrices , prove $Bv = \Lambda v$

$A,B$ are $n \times n$ matrices and $AB = BA$ Also, there is an eigenvalue $\Lambda$ in $A$ which its geometric complexity is $1$. Also there is $ v \ne 0 $ that $v$ is an eigenvector of $A$. ...
1
vote
3answers
21 views

How to find orthogonal projection of vector on a subspace?

Well, I have this subspace: $V = \operatorname{span}\left\{ \begin{pmatrix}\frac{1}{3} \\\frac{2}{3} \\\frac{2}{3}\end{pmatrix},\begin{pmatrix}1 \\3 \\4\end{pmatrix}\right\}$ and the vector $v = ...
0
votes
1answer
19 views

Subspace that contains a particular vector

Let $W \in R^4$ be the smallest subspace that contains the vector $(0,1,1,-1)$ and the vectors $ (t,0,t-1,t+2)$ for every $ t \in Z$ Find a base and the equations of $W$ A base of $W$ is: $ W = ...
-2
votes
0answers
34 views

Finding the general solution for a nonhomogeneous linear equations.

Let $\quad {a}_{i},{b}_{i} \left(i=1,2,\dots,n \right)\in \mathbb{K},\mathbb{K}$ is a Field. $$\begin{cases} & \text ...
2
votes
3answers
36 views

What kind of transformation an upper triangular matrix represents

Every matrix represents a linear transformation, but depending on characteristics of the matrix, the linear transformation it represents can be limited to a specific type. For example, an orthogonal ...
1
vote
0answers
25 views

2x2 Eigenspaces

A couple of questions regarding the solution above. In iii) I understand how to work out $[T]_B$ but i don't understand why this is equal to $A$. In the solution to Q4 when working out the ...
1
vote
1answer
69 views

First Order Logic Consistency Big Problem

as i read some tutorial material on First Order Logic, i deduce that the following formula was consistent in FOL except the third one. am i right? i have doubt about the first one. any idea? thanks to ...
1
vote
1answer
57 views

Gram-Schmidt: Do the sets have some sort of order?

I'm learning about the Gram-Schmidt process: I have some subspace basis $A$ with three vectors: $$A = \{a_1,a_2,a_3\}.$$ Based on it, we will create an orthonormal basis $B$ with three vectors, ...
3
votes
2answers
30 views

Rational quadratic forms

The quadratic form $$10x^2+20y^2+2z^2+4xy-6xz+8yz$$ can be written as $x^TAx$, where A = [ [10,2,-3] , [2,20,4] , [-3,4,2] ] Using diagonalization, this can be written in the form ...
2
votes
3answers
49 views

Where are linear equations with large number of variables used?

Do weather prediction / financial models or missile / rocket trajectory prediction model use these equations? What method or algorithm is used for the same?
2
votes
1answer
52 views

Neighbour Points in N-Dimensional Space

if you got a integer point in the n-dimensional space how many neighbor integer points does it have? 1D you have 2 2D you have 8 3D you have 26 i came to the formula $$n_i = 2*(n_{i-1}+1)+n_{i-1} ...
1
vote
1answer
35 views

Index of a function and a gradient flow

We know index of function $F:\mathbb{R}^n\to\mathbb{R}$ at critical point $x_0\in\mathbb{R}^n$ is the number of negative eigen values of Hessian matrix $DF^2(x_0)$. ...
-1
votes
0answers
28 views

Order of eigenvectors in jblas?

I am using jblas to compute eigenvectors of a double symmetric matrix. Using symmetricEigenvectors(myMatrix)[0], I can get a matrix which columns are the eigenvectors of my matrix. However I need them ...
2
votes
1answer
55 views

Linear dual of vector fields

Suppose that $M$ is a smooth manifold and $\mathfrak{X}(M)$ is the set of smooth vector fields on $M$. There are basically two different linear structures on $\mathfrak{X}(M)$: 1.) $\mathfrak{X}(M)$ ...
0
votes
1answer
36 views

Show that $\lambda_1 = \min \{ Q(u) \mid \|u\| = 1 \}$ and $\lambda_m = \max \{ Q(u) \mid \|u\| = 1 \}$

Let $V$ a vector space over $K$ and $Q(u) = \langle u, Tu \rangle$ a quadratic form. $T$ is a symmetric operator. The eigenvalues of $T$ are sorted by size $\lambda_1 < \dots < \lambda_m$. How ...
-1
votes
1answer
41 views

Binomial Coefficients form Basis for Rational Polynomials

How would we show that the polynomials $c_n(x):=\dbinom{x}{n}$ form a basis for $\mathbb{Q}[x]$?
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1answer
36 views

Writing $T:V \rightarrow \mathbb{F}$ as an inner product.

Let $V$ be a finite dimensional vector inner product space over $\mathbb{F}$, and let $g:V \rightarrow \mathbb{F}$ be a linear transformation. Then there exists a unique vector $y \in V$ such that ...