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 ...

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1answer
15 views

Proofs involving orthonormal basis

Suppose that $V$ is an inner product space. (a) Show that if $\{e_1, . . . , e_n\}$ is an orthonormal basis for $V$ , then $$||v||^2=\sum_{i=1}^{n}|\langle v|e_i\rangle|^2\quad \quad \text{for every ...
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0answers
10 views

Unique linear combination in matrix with skew-symmetric condition

Let $A$ be an $n\times n$ matrix with real entries such that the numbers in each column sum to $0$, and $a_{ij}\in\{0,1\}$ for all $i\neq j$, and $a_{ij}=0\leftrightarrow a_{ji}=1$ for all $i\neq j$. ...
0
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1answer
35 views

Spans of Orthogonal complements

Let $A$ be the matrix $$ \begin{pmatrix} 1 & 1 & -1&-1 \\ 1 & 2 & -2 & 1 \\ \end{pmatrix} ,$$ let $W$ = ker $A$ and let $W^\bot$ be the ...
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3answers
26 views

Im(AB) $\subset$ Im A

Let A,B be linear operators over $\mathbb{R}^n$, where $n > 2$. Which of the following statements is correct? Im(AB) $\subset$ Im A Im(AB) $\subset$ Im B Im(AB) $\supset$ Im A ...
1
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1answer
45 views

Proving that the matrix exponential map is surjective onto the general linear group

Let $M_n(\mathbb{F})$ be the set of all $n\times n$ with entries in $\mathbb{F}$ and let $\exp:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be defined by $$ \exp(A)=\sum_{k=0}^{\infty}\frac{A^k}{k!},$$ for ...
2
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1answer
16 views

Unique linear combination in matrix with column sum $0$?

Let $A$ be an $n\times n$ matrix with real entries such that the numbers in each column sum to $0$, and all diagonal entries are non-zero. So, $A$ is non-invertible, and some linear combination of ...
2
votes
1answer
36 views

Is there an easier way to find the inverse of a 3x3 matrix?

I know the normal process is to do row operations to transform the matrix to get the identity matrix and then apply the same row operations in the identity matrix to get the inverse. But this process ...
1
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1answer
18 views

Finding eigenvectors through triangularization

I have an exam tomorrow and am working through notes. We derived the following stochastic matrix: $$P=\left[ \begin{matrix} 0.8 & 0.5 & 0 & 0\\ 0.2 & 0.5 & 0 & 0 \\ 0 & 0 ...
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1answer
17 views

Solution of system of equations in prime fields

In 'Algebra', Artin writes that the system of equation: $$8x+3y = 3$$ $$2x+6y = -1$$ have no solutions in $\mathbb{F}_2$ and $\mathbb{F}_3$ as the determinant (of the coefficient matrix) evaluates ...
1
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1answer
44 views

Derivative of a $1 \times 1$ matrix [on hold]

What is the derivative of a $1 \times 1$ matrix? Would it just be the derivative of the element inside?
2
votes
1answer
30 views

Find orthogonal Q given eigenvalue and eigenvector?

Given some upper Hessenberg matrix $H \in R^{n \text{x} n}$, i know how to find an orthogonal matrix which is a product of Givens rotations such that $P^THP$ is also upper Hessenberg, but I'm not sure ...
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0answers
35 views

Help: Studying A-Level Mathematics [on hold]

Although I am a latecomer at the age of 21 years of age, I have enrolled in self taught mathematics A-level with "Edexcel" both mathematics & further mathematics. I am in need of help with ...
2
votes
0answers
40 views

Why is unit circle not sufficient to bound the partial sums?

I want to find vectors $\textbf{v}_1, \dots,\textbf{v}_n$ in $\mathbb{R}^2$ with that $\sum_{i=1}^n\textbf{v}_i=\textbf{0}$ and $\Vert \textbf{v}_i\Vert\leq 1$ for all $i=1,\dots,n$, such that for ...
1
vote
1answer
27 views

Proof that an $n \times n$ matrix is positive definite iff all of its eigenvalues are positive

I am trying to prove that an $n \times n$ matrix is positive definite iff all of its eigenvalues are positive. I know that if $\lambda$ is an eigenvalue then: $Ax = \lambda x$ for eigenvalues ...
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0answers
23 views

Draw the parallelogram spanned by the vectors.

I am trying to figure out how to solve the following generic problem: Draw the parallelogram spanned by the vectors $\begin{bmatrix} q \\ r \end{bmatrix}$ and $\begin{bmatrix} s \\ t \end{bmatrix}$. ...
1
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1answer
20 views

Is the set $V$ = { $([t], [g], [t], [j]): t,g,j∈$Z$,[2t+j] = [0]$} a subspace of vector space $(\mathbb Z_3)^4$?

Is the set $V$ = { $([t], [g], [t], [j]): t,g,j∈Z,[2t+j] = [0]$} a subspace of vector space $(\mathbb Z_3)^4$? I am inclined to think that it is a subspace. However, I cannot find any basis for the ...
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0answers
18 views

Co ordinate independent linear algebra over graphs

It is frequently said that Linear algebra is not correct until it is coordinate free or something to that effect and indeed, almost all the major results can be stated without picking a basis. ...
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0answers
8 views

Find the derivative of quadratic form - Product Rule for Matrix Derivatives?

I am trying to find the derivative of the following expression \begin{eqnarray} b' y' Z ((b' \otimes Z') \Sigma (b \otimes Z))^{-1} Z'yb \end{eqnarray} where all matrices are real and b is $p$ by 1, y ...
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2answers
29 views

Linear Applied Algebra | Verify the vectors

Question: My response: Math has never been a strength, particularly proofs, so I would appreciate any and every help. I am just not sure if I am following a proper procedure for the above ...
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1answer
36 views

Applied Linear Algebra | Linear Dependent Matrix

Question: My response: Am I solving the above question correctly? Or am I on the wrong path? Thank you for your help.
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2answers
26 views

Applied Linear Algebra |

Question: Determine whether or not any column in the matrix is a linear combination of other columns. Provide a general method for answering the same question for any n x n matrix A. My response: ...
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0answers
20 views

Applied Linear Algebra | Linear Dependent Matrix [duplicate]

My response: Am I correct in my approach? Thank you for your help.
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1answer
44 views

Prove that the set $\{ x^2 + 4x -3, 2x^2 +x + 5, 7x - 11\} $ does not span $\textit{P}_2$.

Could someone please explain how to prove this? Also, why is it that we must create a set of coefficients for every polynomial contained in S in order to prove it and why is rank so significant? ...
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1answer
31 views

Applied Linear Algebra | Prove the intersection of two subspaces

Question: Determine whether or not any column in the matrix is a linear combination of other columns. Provide a general method for answering the same question for any n x n matrix A. My response: ...
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2answers
22 views

When is the transpose of a square unitary matrix also unitary?

If I have a unitary square matrix $U$ ie. $U^{\dagger}U=I$ ( $^\dagger$ stands for complex conjugate and transpose ), then for what cases is $U^{T}$ also unitary. One simple case I can think of is ...
2
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1answer
24 views

Exercise on linear maps with a nilpotent one

sorry for asking to help me with this trivial problem. Unfortunately I'm in a very bad shape with linear algebra, being this the fourth exercise I'm not able to solve. I need some suggestion. ...
0
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1answer
21 views

Relationship between eigenvectors of two matrices

Suppose I have matrix $A \in R^{2n \text{x} 2n} $ given by $X^{-1} diag(W - iY, W + iY) X$ and matrix $B \in C^{n \text{x} n}$ and $B = W + iY$. Let $v$ be an eigenvector of $A$. How can I relate ...
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2answers
37 views

Calculate matrix $X$ in expression $X + B = (A-B)X$

I have to calculate matrix $X$ in expression $X + B = (A-B)X$. $$ A=\left[ \begin{array} k1 & -2 & 3\\ 2 & 4 &0\\ -1 & 2 & 1\\ \end{array} ...
0
votes
1answer
25 views

Looking for help on orthogonal lemma

Hello I am trying to understand the proof in my notes and I don't get it. I am looking for someone to show me the proof or if possible tell me the name of this theorem so I can look it up. This is not ...
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votes
2answers
19 views

finding dimension of a subspace [on hold]

Find the dimension and basis of the following vector space over a field $\mathbb K$: $V$ is the set of all vectors $(a,b,c)$ in $\mathbb R^3$ with $a+2b-2c=0$, $\mathbb k= \mathbb R$. I can't see how ...
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0answers
30 views

Proving that eigenvalues are positive iff $det(A_k)> 0$ for all $k = 1, …, n$ for a real symmetric matrix $A$

I am trying to prove that eigenvalues of $A$ are positive iff $det(A_k)> 0$ for all $k = 1, ..., n$ for a real symmetric matrix $A$ where $A_k$ is the $k \times k$ matrix obtained by deleting the ...
3
votes
1answer
51 views

How to curve fit an unknown function?

I have data which can be described by $y=f(x,z)$ where $z$ varies from 170 ~ 154. Now values given by $ks$ are known sample values that equals value given in the table header, $uks$ are unknown ...
6
votes
1answer
43 views

Show that $Y$ is invertible

Let X be a $40\times40$ matrix such that $X^3 = 2I$. I want to show that $Y= X^2 -2X + 2I$ is invertible as well. I tried working with the equations to see if I can get Y as a product of matrices ...
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votes
0answers
17 views

Dot product in bilinear form (Euclidean space) [on hold]

Find $a$, a real number such as $$ B((x,y),(x',y'))=xx'+2xy'+2x'y+ayy'$$ is a dot product.
1
vote
1answer
48 views

Find spectrum for matrix $A$ [duplicate]

Let $A = \left[ \begin{array}{*{20}{c}} 0&b&0&0&0&0\\ c&0&b&0&0&0\\ 0&c&0&b&0&0\\ 0&0&c&0&b&0\\ ...
3
votes
1answer
74 views

Find x,y & z (xyz+xyz=zyx)

I saw this problem the other day at work and found it pretty interesting: $$xyz + xyz = zyx$$ Find $x, y, z$ and the base(s) which this is true. Note that $x,y,z$ are simply digits concatenated, ...
0
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0answers
37 views

How to find out if this vector system of functions is linearly independent?

For example, i have these functions in the vector space $\mathbb R^{\mathbb R}$: $x^2-x+3$, $2x^2+x$, $2x-4$ And I have to determine if they are linearly independent, how should I solve this ...
2
votes
1answer
29 views

If $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$ are in $d$-general position, then they are in $1$-general position.

Let $\mathcal{L}_{d}^{n}$ be the $\binom{d+n}{n}-1$ dimensional projective space of hypersurfaces of degree $d$ in $\mathbb{P}^{n}$ and $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$. We denote by ...
0
votes
1answer
33 views

Find matrix representation of transformation

Given two lines $l_1:y=x-3$ and $l_2:x=1$ find matrix representation of transformation $f$(in standard base) which switch lines each others and find all invariant lines of $f$ My attpempt is to ...
1
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2answers
40 views

Prove linear transformation

I'm working on linear transformation trying to answer : Let $E$ and $F$ be two vector spaces on $\mathbb{K}$ and $L:E \rightarrow F$ a function. The graph of $L$ is $\mathbb{G}(L)=\{(x,y) \ \in \ ...
0
votes
1answer
50 views

Show convergence of Power method

Given a symmetric positive definite matrix $A_0 \in R^{n \text{x} n}$ with Cholesky decomposition $A_0 = LL^T$. How can I show that $A_k$ converges to $diag(\lambda_1, ..., \lambda_n)$ where $A_k$ is ...
0
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1answer
15 views

proof involving field

let $A$ and $B$ be elements of a field, and suppose that $AB=0$. Prove that at least one of $A$ and $B$ must be equal to $0$. Here is my answer: $AB=0$, $AB=A.0$, $AB-A.0=0$, $A(B-0)=0$, hence either ...
1
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4answers
27 views

Basis of a $2\times2$ matrix

How would I find the basis for an arbitrary matrix W such that: $$ W =\left\{ \begin{pmatrix} a & b \\ c & a +b +c\end{pmatrix} \ \big| \ \ a ,b ,c \in \mathbb{R} \right\} $$
4
votes
1answer
46 views

Is $\text{rank} (AA^*)=\text{rank}(A)$ for all nonsquare matrices? [duplicate]

If $A$ is a $m\times n$ type matrix with $m\geq n$ then $$ rank (A^*A)=rank (A). $$ Is maybe also true in general that $$ rank (A^*A)=rank (A) ? $$ Thanks Edit. My question is different from the ...
-1
votes
0answers
21 views

How do I fit this piece of code on one line in Latex [migrated]

I am trying to have three matrices on one line as i) A=, ii) B= and ii) C=. I tried \nopagebreak, \noindent just after item. Instead I always get the Roman numeral on one line, a comma on the next ...
1
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0answers
20 views

Skew-symmetric non-degenerate bilinear form

If we do symplectic linear algebra on a finite-dimensional vector space $V$, then what does $$\omega(v,w) \neq 0$$ or $$\omega(v,w) = 0$$ actually tell us about the vectors $v,w$? ($\omega$ is the ...
0
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2answers
30 views

Proove L is a linear transformation

I'm working on linear transformation and trying to answer : Let E and F be two vector-spaces on $\mathbb{C}$ and $L:E \rightarrow F$ an application such as : $\forall u,v \in \ E, L(u+v)=L(u)+L(v) $ ...
0
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0answers
35 views

Equating eigenvalues of Hermitian matrix and correlating symmetric/antisymmetric matricies

I have a matrix $AH$ which is created by adding $AS$ and $i*AA$, which are the symmetric and antisymmetric components of the real matrix $A$ So $AS=(A+A')/2$ $AA=(A-A')/2$ $AH=AS+i*AA$ AH has ...
0
votes
1answer
16 views

matrix sampling and its rank preservation

Assuming matrix $X\in R^{m\times n}$ is row orthogonal of rank $m$. Then, if I construct a new matrix $Y\in R^{m\times t}$, whose columns are directly sampled from $X$ with or without replacement ...
0
votes
2answers
42 views

Prove that the output of the function equals the determinant

Let $δ$ : $M_{2×2}$($F$) $→$ $F$ be a function with the following three properties. ($i$) $δ$ is a linear function of each row of the matrix when the other row is held fixed. ($ii$) If the two rows ...