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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Proof for $\mathbf{M}$ unitary if $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$

Let $\mathbf{M} \in \mathbb{F}^{n, n}$. Then $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$($\mathbf{v} \in \mathbb{F}^n$) implies that $\mathbf{M}$ is unitary. My question is, how to prove this ...
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5answers
40 views

Is it possible to solve for $m$ in a linear equation without knowing $b$?

Suppose you know certain points on a line say $(5,2)$ up to $(8,10)$ but you don't know exactly where the $y$ intercept would be being somewhere down there at like $-25$ area. How would you solve for ...
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1answer
50 views

Which matrices diagonalizes a diagonal matrix? [on hold]

I think the answer is the set of all diagonal matrices but I am not sure. Can anyone give the answer with a proof?
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4answers
106 views

Do row operations change the column space of a matrix?

I know that (i) row operations do not change the row space (ii) column operations do not change the column space and (iii) row rank = column rank (but this is sort of unrelated, I think). But, ...
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2answers
58 views

Prove that similar matrices have the same nullity.

How do I approach this? I'm assuming it might have something to do with $B = P^{-1}AP$.
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0answers
46 views

Practice Exam question need help! [on hold]

For vectors $f,g \in C[-\pi,\pi]$, we use the inner product $\langle f,g \rangle = \displaystyle \int_{-\pi}^{\pi} f(x)g(x)\,dx$. Then, $S=\{1/(2\pi)^{1/2},\sin(x)/\pi^{1/2}\}$ is an orthonormal set ...
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2answers
45 views

Finding Eigenvalue det(λI - A);

I want to know if what I'm doing to derive equation (2) from (M2) is correct or not; usually, before moving onto the next row in Guass-Jordan elimination we turn a_11 into a leading one or whatever ...
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1answer
30 views

Problem about dual of $W = V \oplus V'$

Let $V$ by finite dimensional, let $W = V \oplus V'$, and prove that the correspondence $(x,y) \rightarrow (y,x)$ is an isomorphism between $W$ and $W'$. (The direct sum is defined as the set of ...
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2answers
34 views

Help With Finding A Basis

I came up to the following matrix: $$\begin{pmatrix} 3 & 1& 3& -4\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ \end{pmatrix}$$ I know that ...
2
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2answers
39 views

Do linear operators that map one space into a different space have a Jordan canonical form?

I know that this answer is most likely "yes", and that, in the setting of matrices, all matrices are similar to its Jordan form, which is unique (up to the ordering of the Jordan blocks.) But what ...
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3answers
17 views

Finding Rank And Eignvalues Of Vectors Multiplication

Let $v=(3,1,3,-4)$ and $A=v^tv$, Find: the rank of $A$ $Null(A)$ eigenvectors and eigenvalues Is there a way to approach this without finding $A$ explicitly?
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1answer
12 views

Question about row operations and row-echelon form,

If I have a matrix, with, say, the first two columns consisting of all zeroes, then is the first entry of the third column, which is non-zero, my first pivot variable, so that when solving Ax=b, for ...
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2answers
19 views

Distributive property of scalar multiplication over scalar addition

I need help with a simple proof for the distributive property of scalar multiplication over scalar addition. Help with proving this definition: $(r + s) X = rX + rY$ I have to prove the truth of the ...
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1answer
22 views

how to calculate the projection of a vector onto a closed convex set? [duplicate]

suppose we have a vector $x \in \mathbb{R}^{n}$, and a closed convex set $C \in \mathbb{R}^{n}$. $C =\{x|Ax=b\}$ how to calculate the vector $y \in \mathbb{R}^{n}$, which is the projection of $x$ ...
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1answer
24 views

Proving space of skew-symmetric matrices is orthogonal complement of symmetric matrices

Problem: Prove that $\left\{ A \in \mathbb{R}^{n \times n} \mid A \text{ is symmetric}\right\}^{\bot} = \left\{ A \in \mathbb{R}^{n \times n} \mid A \ \text{is skew-symmetric}\right\}$ with $\langle ...
3
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2answers
32 views

Eigenvalues of Matrix Product.

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their product? What about the special case when one of these matrices is a diagonal (positive) matrix? I ...
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3answers
54 views

Help with proof: $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$

The question is: If $A,B$ are any $m\times n$ matrices, prove that $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$. ($\mathrm{rank}(A)$ is the dimension of the column space of $A$, ...
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3answers
34 views

Eigenspace and $\ker(T)$

It seems like eigenspace and $\ker(T)$ are strongly connected, I have thought about some properties and I would like to make sure I got it right. for all matrix/transformation there is an Eigenspace ...
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0answers
10 views

reoder basis vectors to get 'more diagonal' representation of NxM matrix

I am trying to reorder the basis vectors of an NxM matrix in a way that leads to a representation with 'as much weight close to the diagonal as possible'. I would be happy, if instead of a matrix ...
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2answers
25 views

Finding orthonormal basis for a subspace $W$ of the Euclidean space $\mathbb{R}^3$.

Problem: Let $\mathbb{R}^3$ be an Euclidean space. Find an orthonormal basis for the subspace $W$ defined as $x + 2y-z = 0$. Attempt at solution: So this is a plane in $\mathbb{R}^3$, so I guess I ...
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2answers
41 views

Every vector space is isomorphic to the set of all finitely nonzero functions on some set

I am trying to prove the statement in the title, that Every vector space is isomorphic to the set of all finitely nonzero functions on some set. A finitely nonzero function from $X \rightarrow ...
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1answer
32 views

Using the Gram -schmidt procedure to find the orthonormal set (Linear Algebra)

(a) Construct an orthonormal basis of the space $R^3$ satisfying the requirment of the Gram-Schmidt prodcure from the basis $v_{1}=(-3,4,0)$ , $v_{2}=(5,10,-24)$ , $v_{3}=(0,0,1)$ (b) Given that ...
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2answers
78 views

Possible to express the diagonal matrix $D=\tiny\left(\begin{matrix}1&0&0\\0&2&0\\0&0&3\end{matrix}\right)$ as function of $3\times3$ identity matrix?

Is possible to express the diagonal matrix $$D=\left(\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}\right)$$ as function of $3\times3$ identity matrix? For ...
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0answers
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A qustion in matrix polynomial [on hold]

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
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3answers
23 views

Size of a triangle using determinant [duplicate]

find the size of a triangle using (determinant) with the following points: $(x_1,y_1)=(1,-2)$ $(x_2,y_2)=(-4,-2)$ $(x_3,y_3)=(-5,-1)$ How should I place those points in the ...
2
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2answers
44 views

Prove there's a unitary linear operator

Let $u, v\in V$, where $V$ is a finite dimensional vector-space, such that $\|u\|=\|v\|$. Prove there's a unitary linear operator such that $T(u) = v$ So if there's such unitary linear operator, it ...
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1answer
16 views

How to prove this “local invertibility” theorem for bounded linear operators?

The theorem states that, suppose $X,Y$ are complete normed vector spaces, if $\mathscr A_0\in \mathscr L(X;Y)$ is invertible (i.e., $\exists \mathscr A_0^{-1}\in\mathscr L(Y;X)$ s.t. $(\mathscr ...
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2answers
57 views

An inequality for the dimension of the sum of subspaces

The answer with the most of upvotes on MO is this answer on $\dim(U+V+W)$. Question: 1. Is it nonetheless true that every three vector subspaces $U$, $V$ and $W$ of a vector space $M$ satisfy $$ ...
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1answer
15 views

if $E,F$, two bases are orthonormal then $T$ is unitary.

Let $T:V\to V$ and two bases of $V$: $E = \{v_1, \ldots, v_n \}$ and $F = \{T(v_1), \ldots, T(v_n)\}$. Prove: $E,F$ are orthonormal implies $T$ is unitary. So basically we want to prove ...
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0answers
11 views

Adjugate matrix-Almost Inverase

I Came across the following statement: "If the inverse of the original matrix exists, the adjoint is "almost" that inverse". What does it mean? Moreover, intuitivly, what does the property ...
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2answers
40 views

Normalizing a basis

Let the basis $B = \{1,x,x^2\}$ which is orthogonal. Now, I've seen the following: $$\|1\| = \sqrt {\langle 1,1\rangle} = \sqrt {4\cdot 1\cdot 1} = 2 $$ $$\|x\| = \sqrt {\langle x,x\rangle} = ...
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2answers
238 views

If two matrices have the same trace and determinant, do their powers have the same trace?

Let $A,B$ be two $2 \times 2$ matrices over some finite field $\mathbb{F}_q$, such that they have the same trace and determinant. Does this imply that tr $A^k$ = tr $B^k$ for any integer $k$? I've ...
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0answers
43 views

How can I use the Bullet-Physics's ray-cast normal to calculate angles for a object to lay on a surface?

[Give the normal of a surface in XYZ format, how do I calculate rotations (also in XYZ format) needed to set an object parallel to the surface?] I have a collision library that uses the bullet ...
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2answers
24 views

Proving the orthogonality of an inner product space (Linear Algebra)

Prove that any orthogonal set $S$ consisting of non zero vectors is linearly independent. My try By contradiction we assume that the orthogonal set $S$ consisting of non zero vectors is linearly ...
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0answers
35 views

Calculation of $trace(L^THL)$, L is lower triangular, H is symmetric.

I am working on a problem where I had to find the following expression: $$ l = Tr({P'HP})$$ I already modified my model formulation using cholesky decomposition for PSD matrices and came up with ...
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3answers
45 views

Finding dimension of subspace

I know that any polynomial in subspace $W$ must have $(x-1)$ as factor so that $p(1)=0$ But I don't understand how $p'(2)=0$ can be incorporated. Thankful for any kind of help.
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1answer
21 views

How to denote a tensor in terms of matrices product?

How to write a tensor in terms of a product of matrices? For example, I have $a \times b$ matrix $F$, and I want to create a 3D $a \times a \times a$ tensor $T$, where $T_{i,j,k} = \sum_{m=1}^{b} ...
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1answer
48 views

A result about commuting matrices in $ M(n, \mathbb{C} ) $

Let $ A $ be a matrix in $ M(n, \mathbb{C} ) $ and let $ A^{*} $ be its Hermitian adjoint. Suppose that the matrices $ A $ and $ AA^{*}-A^{*}A $ commute. Show that $ AA^{*} = A^{*}A $. Here is a ...
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2answers
20 views

Eigenvalues of a companion matrix

I've been tasked with the following: Show that the companion matrix $C(p)$ of $p(x) = x^2 + ax + b$ has characteristic polynomial $\lambda^2 + a\lambda + b$. Show that if $\lambda$ is an ...
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0answers
67 views

How coefficients from finite field can form ring of polynomials?

Let us consider a graph $G(V,E),$ where $V$ is the set of nodes and $E$ is the set of edges. $\mathbf{x}=[X_1,\ldots,X_r]$ are symbols multicast by source to $|T|$ sink nodes. Symbols are from ...
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1answer
63 views

Determinant of Matrix combination

I have got a big Matrix with the structure $$ A=\begin{pmatrix} A_1 & A_2 \\ A_2^{T} & A_4 \\ \end{pmatrix} $$ with $A_{1}$ equals a Matrix full of zeros. $A_{2}$,$A_{3}$ and $A_{4}$ can ...
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1answer
38 views

Comparing $\text{tr}(A^{-1})$ and $\text{tr}(A(B+A)^{-2})$ for pd $A$ and psd $B$

Suppose that $A$ is positive definite and $B$ positive semidefinite, both with dimension $n\times n$. Is there some inequality between $$ \text{tr}(A^{-1})\quad\text{and}\quad\text{tr}(A(B+A)^{-2})? ...
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0answers
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Csir net question [on hold]

let V be the space of twice differentiable functions on R satisfying f double dash-2f single dash+f=0. define T from V to R(square) by T(f)=(fdash(0),f(0)). Then T is_______ 1) 1-1 and onto 2) 1-1 but ...
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0answers
31 views

How do I interpret “K2 mod K1”?

I've been doing a bit of work these past couple of nights on computing cyclic subspace decompositions, finding cyclic bases, and then computing the Jordan canonical form of matrices. My question is: ...
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1answer
61 views

Should I use set notation or list notation when writing out a basis of vectors?

I think in Sheldon Axler's Linear Algebra Done Right, he makes a comment about why the technically correct way is to write vectors in lists, such as $(v_1, ... v_n)$, while many books use set ...
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2answers
43 views

Find a plane with distance $3$ from $3x-y-z = 0$

I need to find a plane such that its distance from the plane $3x-y-z = 0$ is $3$. Since distance is defined only for parallel planes, I already know that they have to be parallel, and then, the ...
2
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2answers
21 views

Confusion between spectral radius of matrix and spectral radius of the operator

The adjacency matrix $A(G)$ of an infinite undirected graph $G$ is considered as a bounded self-adjoint linear operator $A$ on the Hilbert Space $l^2(G)$ (last section of ...
3
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1answer
46 views

$6$ eigenvalues of a $4\times4$-matrix?

I am struggling with determining the eigenvalues of the following (symmetric) matrix: $$ A =\begin{pmatrix} 2 & 1 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 ...
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0answers
18 views

Asymptotic series of a matrix-valued function.

Consider the following matrix $$f(\lambda)=\left( \frac{\lambda-1}{\lambda + 1} \right)^{\nu \sigma_3} \ \ \ \lambda \in \mathbb{C} \setminus [-1,1]$$ where $\sigma_3=\begin{pmatrix} 1 & 0 \\ 0 ...
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1answer
25 views

Where did I go wrong with the Gram-Schmidt orthogonalisation process?

Problem: Let $\alpha = \left\{(1,2,0), (1,0,1), (2,3,1)\right\}$ be a basis vor $\mathbb{R}^3$. Apply the Gram-Schmidt orthogonalisation process to turn $\alpha$ into an orthonormal basis for ...