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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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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1answer
15 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
27 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
22 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
10 views

Applied Linear Algebra | Linear Dependent Matrix

My response: Am I correct in my approach? Thank you for your help.
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1answer
35 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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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
20 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 ...
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1answer
23 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. ...
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1answer
19 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
33 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} ...
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1answer
22 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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2answers
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finding dimension of a subspace

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
23 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 ...
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0answers
25 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
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1answer
31 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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16 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.
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1answer
46 views

Find spectrum for matrix $A$

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\\ ...
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1answer
62 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, ...
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0answers
35 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 ...
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1answer
18 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 ...
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2answers
27 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 ...
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2answers
33 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 \ ...
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0answers
25 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 ...
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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 ...
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4answers
25 views

Basis of a $2\times2$ matrix

How would I find the basis for an arbitrary matrix W such that: $$ W =\left\{ \begin{pmatrix} a & b \\ c & a +b +c\end{pmatrix} \ \big| \ \ a ,b ,c \in \mathbb{R} \right\} $$
3
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1answer
41 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 ...
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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 ...
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0answers
15 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 ...
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2answers
24 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) $ ...
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0answers
24 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 ...
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1answer
14 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 ...
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2answers
37 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 ...
2
votes
4answers
81 views

Logic behind finding a $ (2 \times 2) $-matrix $ A $ such that $ A^{2} = - \mathsf{I} $.

I know the following matrix "$A$" results in the negative identity matrix when you take $A*A$ (same for $B*B$, where $B=-A$): $$A=\begin{pmatrix}0 & -1\cr 1 & 0\end{pmatrix}$$ However, I am ...
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1answer
37 views

Gauss Method to show [on hold]

Could you please give me the way to solve this problem Using Gauss method to show if $x ≠ y + 1$ then $$ \sum_{i=0}^n (x-y)^i = \frac{(x-y)^{n+1}-1}{x-y-1}. $$
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2answers
52 views

how to convert log(x) into linear form? [on hold]

I have simple function which is non-linear like log(x) I want to convert it into linear function. Anyone could help out? Thanks
0
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1answer
26 views

Reflection matrix in $ \mathbb{R}^{3} $.

I need help in understanding how they got the transformation matrix $ Q_{L} $ from Theorem 2 and $ P_{M} $ at the bottom of the page. They skipped some steps and I find it confusing. Any help would ...
1
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1answer
18 views

Can independence of a system and a vector be establish if there is only cross-indepedence?

Say that I have the following linear system: $$[A a'] \begin{bmatrix} x \\ x' \\ \end{bmatrix} =Ax + a'x' $$ I want to know when this system is zero if and only if ...
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1answer
23 views

Eigenfunctions and Eigenvalues of a Linear Operator

For a math project on Schroedinger's equation I and a partner are working on we need to find eigenfunctions and eigenvalues that satisfy $L\phi_n = \lambda_n\phi_n$, where $L$ is defined as $L\psi = ...
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2answers
20 views

Algebra verbal find the amount of sold items

Hey I am having an exam tomorrow, so I looked up at some verbal algebra questions, and found one that I could not solve, because I don't really understand how would I do this. The question is like ...
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0answers
17 views

About lower bounds on the size of irreducible representations of subgroups of symmetric groups.

Is there a subgroup $G_n$ of $S_n$ (one $G_n$ for each $S_n$) increasing in size such that their permutation representations are such that the smallest non-trivial irreducible size in them is ...
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0answers
17 views

Characterise the set of inner products which are preserved by a given automorphism?

Let $V$ be a finite dimensional vector space. Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry. (You can ...
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1answer
24 views

Finding eigenvalue and eigenvectors of a matrix containing an imaginary number

How do you solve for the eigenvalues given the matrix? \begin{matrix} i & -2 \\ 1 & 0 \\ \end{matrix} I know how to get the characteristic polynomial Ca(X); X^2 - ...
1
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2answers
19 views

Approximating length of a curved line based on Begining and End points of line

I have two points, a known distance apart. At each of these points I have a sensor that gives me flow speed and direction. I originally assumed the flow path between the first point and second point ...
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23 views

Change of eigenvectors by change of coulmn vectors.

This question is an extension to the question in the link: Change the matrix by multiplying one column by a number. It is understood now that if we change a positive definite matrix A to B by ...
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0answers
20 views

How to find spectral radius of ${0,1}$ and ${0,1,-1}$ matrices?

[this is kind of a continuation of this question ] It seems that the following is true, Among $n=3$ dimension symmetric matrices over $\{0,1\}$ which have $d=7$ ones the maximum spectral radius is ...
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1answer
18 views

linear algebra characteristic values [on hold]

Let $T$ be the linear operator on $\mathbb{R}^4$ which is represented in the standard ordered basis by the matrix $$ \left( \begin{matrix} 0 & 0 & 0 & 0 \\ a & 0 ...
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2answers
73 views

Matrices where A^2 = A

I have a feeling that the only invertible matrix - A . that when it squared A^2 is still A , is the Identity matrix . Am I right? and if so , could anybody show me the proof?
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3answers
41 views

Change the matrix by multiplying one column by a number.

Consider a positive definite matrix A. We can think of the matrix as a linear transformation. Now supopse we get matrix B by multiplying only one column of A by a number. Is there a geometric relation ...
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0answers
11 views

What scaled version of vector to use in QR-factorization when vector subtraction is involved

Im trying to figure out if I understand the conceptual basics. All the time you see that vectors are scaled down/up for readability or for simplifying future calculations with that same vector. As ...