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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How to construct a vector space and compute basis?

My professor demonstrated that in vector calculus that you can construct basis vectors for one, two, and three forms using the vectors $dx$, $dx$ and $dy$, as well as $dx \wedge dy$, $dy \wedge dz$, ...
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32 views

Do I have the correct mental map for adjoint operators for inner product spaces?

Let $X$, $Y$ be finite dimensional inner product spaces, let $A: X \to Y$ be a linear operator, let $A^*: Y \to X$ be the adjoint operator to the linear operator, defined using $<y, Ax>_Y = ...
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57 views

If $A$ and $B$ are arbitrary $m\times n$ matrices, show that $^t(A+B)= {}^tA+{}^tB$?

I'm reading Lang's: Introduction to Linear Algebra. There is this exercise: If $A$ and $B$ are arbitrary $m\times n$ matrices, show that $^t(A+B)= {}^tA+{}^tB$ I did the following: ...
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31 views

Regarding the construction of the tensor bundle

Recall the construction of the tangent bundle: we write $$TM = \bigsqcup_{p \in M}T_p M$$ and define it as the prevector bundle with local trivializations $[\gamma] \mapsto (\gamma(0), (x\gamma)'(0))$ ...
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Working in a field.

When I calculate vector spaces, diagonalization of matrices, linear transformations on a field, can I work in $\mathbb R$ and ultimately transform the result to that field? For example if I calculate ...
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72 views

Proving that $\det(A) = 0$ when the columns are linearly dependent

Proposition: Let $A$ be a $(n \times n)$-matrix. If the columns of $A$ are linearly dependent, then $\det(A) = 0$. Attempt at proof: Let $A = (A_1, A_2, \ldots, A_n)$, where each $A_i$ is a column ...
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56 views

Why we define the adjoint operator

Suppose in vector space $A: X\rightarrow Y$ is a linear map, the adjoint operator $A^{'}: Y^{'}\rightarrow X^{'}$ is defined as: $f(Ax)=(A^{'}f)(x)$. As I can understand, the adjoint operator just ...
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28 views

Can we prove existence of Spectral Decomposition from Singular Value Decomposition(SVD)??

The Spectral Decomposition Theorem is, if $A$ is hermitian , then there exists a unitary matrix $U$ such that $$U^{*}AU=D $$ where $D$ is a diagonal matrix. By using the existence of SVD, can we ...
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100 views

Spectral radius of a real, symmetric, positive semi - definite matrix.

While answering a question, the OP made a follow - up question, that I was not able to answer at that moment. However, I came up with an intriguing (at least to me) question. Let ...
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25 views

What does “the activation of a basis” mean?

In the paper Rajat Raina, Alexis Battle, Honglak Lee, Benjamin Packer, Andrew Y. Ng, Self-taught learning: transfer learning from unlabeled data, ICML '07 Proceedings of the 24th international ...
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15 views

How to perturb an adjacecny matrix in order to have the highest increase in spectral radius?

Let's suppose I have a generic directed graph $G$ and it's adjacency matrix $A$. I can add an arc wherever I want in the graph. (i.e. perturb the matrix A changing a single 0 into a 1). Where should ...
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33 views

Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
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70 views

Are all 7-dimensional cross products isomorphic?

Let $\times$ be this 7-dimensional cross product and let $\hspace{.04 in}f$ be a bilinear map on $\mathbb{R}^7$ which satisfies the orthogonality and magnitude conditions. Does there necessarily ...
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61 views

Show that $A^k$ has eigenvalues $\lambda^k$ and eigenvectors $v$.

I want to prove the following statement: Let $A \in \Bbb R^{n\times n}$ with eigenvalues $\lambda$ and eigenvectors $v$. Show that $A^k$ has eigenvalues $\lambda^k$ and eigenvectors $v$. ...
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23 views

Determinant 1 matrix does not change p-adic measures

Let $f:\mathbb Z^d \rightarrow \mathbb Z^d$ be a linear map having determinant 1. Is there an obvious way to see that if $U\subseteq \mathbb Z_p^d$ is a measurable set, then the p-adic measure of $U$ ...
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44 views

Evaluate the norm of a linear operator

Can someone help me on this question. I want to compute the norm of the following operator $$l:\mathbb R^N\longrightarrow \mathcal M_n(\mathbb R); (x_1,\cdots,x_N)\mapsto (x_i - x_j)_{1\leq i,j\leq ...
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53 views

Resultant of two polynomials in two variables

I have two polynomials in two variables. $$f= nx^n+(n-1)x^{n-1}y+(n-2)x^{n-2}y^2+...+xy^{n-1}-c$$ $$g= x^{n-1}y+2x^{n-2}y^2+3x^{n-3}y^3+..+(n-1)xy^{n-1}+ny^n-d$$ Where $c$ and $d$ are some ...
3
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36 views

What do we know about inverses of matrices which are “like” Laplacians of graphs?

Consider the Laplacian $L$ of a bipartite graph. Is there any generic understanding we have about what $1/(z-L)$ looks like? [say $z > \lambda_\max(L)$)] You can consider variations of $L$ like ...
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106 views

Nontrivial normed functional on the bounded functions from $\mathbb R^2$ into $\mathbb R$ invariant by isometries

I am trying to show that there exists a nontrivial normed functional on $\mathbb R^2$ invariant by isometries. That is: If $A$ is any set, let $\mathcal B_{A}=\{f: \mathbb R^2 \rightarrow \mathbb R: ...
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Formula for nth number of the following sequence:

I have two number sequences but have failed to find a formula for the nth term and, also, the formula for the sum of the sequence. First sequence: ...
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37 views

Please help understand dimension of a subset of a subspace

True or False: if $S$ is a subspace of dimension $3$ in $\Bbb R^4$, then there cannot exist a subspace $T$ of $\Bbb R^4$ such that $S$ is a proper subset of $T$ and $T$ is a proper subset of $\Bbb ...
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34 views

Extending Semilinear Transformations over Finite Fields

Suppose $ p $ a prime integer and $ m $ and $ n $ positive integers, where $ m | n $. Let $ \Phi_{m} $ denote the Frobenius automorphism of $ \mathbb{F}_{p^m} $ and $ \Phi_{n} $ the Frobenius ...
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Find the number of distinct real values of $c$ such that $A^2x=cAx$

Let $$A= \begin{pmatrix} 5 & -3 & 0 \\ -3 & 5 & 0 \\ 0 & 0 & 2 \end{pmatrix}$$ and $c$ be a real no. such that $A^2x=cAx$ for some non-zero vector $x$. Then the number of ...
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24 views

Weight spaces of a irreducible representation of $\mathfrak{gl}(n, \mathbb{C})$.

Let $\mathfrak{gl}(n,\mathbb{C})$ be the general linear Lie algebra. Let $\{E_{s,t}\}_{1\leq s,t,\leq n}$ be the standard basis for it. And set its Cartan subalgebra $\mathfrak{h}$ to be ...
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64 views

Find a symmetric matrix B that makes ABC symmetric, A,C known

I have two known matricies $\bf{A} \in \mathbb{R}_{nxm} $, $\bf{C} \in \mathbb{R}_{mxn}$ with $m>n$. I'm trying to find a $\bf{B} \in \mathbb{R}_{mxm}$ that is symmetric and makes $\bf{ABC}$ ...
3
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32 views

Gram-Schmidt in characteristic two?

I was helping someone work on a computing problem with bit vectors that reduced to finding a basis knowing a spanning set, and realized quickly that the Gram-Schmidt process does not work as expected ...
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34 views

A matrix representation for the inverse matrix.

I have the next problem from the textbook: "Methods of Algebriac Geometry in Control Theory by Peter Falb". Problem 1: Show that if $A$ is an $n\times n$ matrix, then $(zI-A)^{-1} = \sum_{j=1}^n ...
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18 views

Find a matrix to represent the mapping of a factor module

I have a problem from my past paper I can't figure the logic to, even after seeing the answers. The question goes 【Q】Let $V=\mathbb{R}[X]_{<4}$ be the vector space of real polynomials of degree ...
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71 views

Angle between two vectors in $\mathbb{R}^n$?

I know that in $\mathbb{R}^2$, the dot product of two vectors $\bf a$ and $\bf b$ is given by $$\mathbf {a} \cdot \mathbf {b} = \mathbf{|a|\,|b|}\cos(\theta), \tag 1$$ where $ | \cdot |$ denotes the ...
3
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65 views

Proof that Determinant is Scale Factor

I've seen a lot of supposed properties of linear transformations that're never proven -- just often repeated. These include: The determinant is the scale factor between the volume of region in your ...
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69 views

Finding a summarizing vector for average angle calculation

Let $L$ and $R$ be two bags of positive vectors such that all vectors have length $k$. Define the distance $d_{avg}$ between the bags as the average pairwise angle between the vectors. Is is possible ...
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52 views

Basis of $\mathbb{F}[[x]]$ over $\mathbb{F}$ without AC

Does the ring of formal power series $\mathbb{F}[[x]]$ as a vector space over $\mathbb{F}$ admit a basis without assuming the Axiom of choice, at least in some special cases of $\mathbb{F}$? I'm ...
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24 views

Convergence of Arnoldi method

I would like to compute the largest real eigenvalue of a matrix in the following form: $$\begin{bmatrix} 0 & I_n \\ P & Q \end{bmatrix},$$ where $I_n$ is the $n \times n$ identity matrix, $P$, ...
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34 views

Which elements of $su(n)$ commute with those of a subalgebra $su(2)$

Given a subalgebra $su(2) \subset su(n)$ , how many generators of $su(n)$ commute with any element in the subalgebra $su(2)$? I know that there are at least $n-2$ elements in $su(n)$ satisfying this ...
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76 views

Prob. 10, Sec. 3.2 in Erwine Kreyszig's “Introductory functional analysis with applications”

Here is Prob. 10 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: ... Let $T \colon X \to X$ be a bounded linear operator on a complex ...
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59 views

Number of solutions of an equation

Fix a vector $x\in\{0,1\}^n$, and let $a$ be a random vector in $\mathbb{Z}^n_q$ for some prime $q$. Consider $y=ax$, and $S=\{x'\mid ax'=y\}$. I want to compute the probability that $\lvert S ...
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17 views

Optimal Matching Distance

I'm stuck on problem II.5.9 from Bhatia's Matrix Analysis. The problem is as follows: Let $\{\lambda_1,\dots,\lambda_n\},\{\mu_1,\dots,\mu_n\}$ by two $n$-tuples of complex numbers. Let $$ ...
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47 views

Decompose a vector space into invariant subspaces?

Consider the following proposition: Suppose $V$ is a finite dimensional vector space over a field $F$, and $K/F$ is a finite Galois extension with Galois group $G$. If $V$ has a $(K,K)$ bimodule ...
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93 views

Does this have a name?

While messing around, I seem to have stumbled upon an interesting family of matrices: $$\mathbb{S} = \bigg\lbrace A\in\mathbb{M}_{n\times n}(\mathbb{R}) : A^{T}A=AA^{T}=\frac{1}{2} (A + ...
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39 views

Forming the graph $G$ from elements of the cut and cycle space, using a weird hint

I'm working through a set of lecture notes on my own, and since there is no class, there are no immediate faculty members available to ask questions to. I've managed to finish most exercises quite ...
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54 views

Relation between $y=Ax_1$ and $y=WAx_2$

I have a question. Is there any relation between the following linear equations? $$y=Ax_1 \ \ \text{ and} \ \ y=WAx_2$$ W is diagonal square invertable matrix, A is an mxn matrix with $n>m$. I ...
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54 views

Eigenvalues of adjugate matrix of a singular matrix

Given a singular matrix $A$, find the eigenvalues of the adjugate matrix of $A$. The same question with $A$ being invertible is trivial since $A\operatorname{adj}A=(\operatorname{adj}A)A=(\det A) ...
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34 views

Iteratively transform vectors to/from the basis of eigenvectors

I have a $n \times n$ banded symmetric matrix $A$ with bandwidth $k \ll n$, so it is quite sparse. The matrix is diagonalizable, $A = V^T \Lambda V$. Generally, $A$ is not positive definite. I need ...
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80 views

How can one prove the existence and uniqueness of solutions to linear differential equations?

It is a theorem (I think) that the equation: $$\mathbf{x}'(t) = A(t)\mathbf{x}(t) + \mathbf{b}(t); \qquad \qquad \mathbf{x}(t_0) = \mathbf{x}_0$$ Has a unique global solution for any matrix ...
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24 views

If $A$ is normal then $A^*=P(A)$ for some polynomial $P$.

Since $A$ is normal it has a diagonal matrix with respect to some orthonormal basis, and the same is true of $A^*$ using the same unitary matrix. We also know that if $\lambda$ is an eigenvalue of $A$ ...
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61 views

Mapping vector spaces over two different fields?

I was having linear algebra class and we have been discussing about a possible group homomorphism that might allow mapping between two vector spaces over two different fields This is also an ...
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38 views

Find all $ p \ge 1 $ for which the Hölder norm $\|\cdot\|_p $ is generated by a scalar product.

Find all $ p \ge 1 $for which the Hölder norm $$ \|x\|_p := \left(\sum^{n}_{i=1} |x_i|^p\right)^{\frac{1}{p}} $$ is generated by a scalar product. We know that norm is generated by a scalar product ...
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23 views

Is this the correct solution involving vector subspaces and basis?

I need to find the basis and hence dimension of a subspace of $\mathbb{R^3}$. 1) $$U=\{(x,y,z):x=2y\}$$ Solution: We have $x=2y \iff y=\frac{x}{2}$ therefore we can write all elements in $U$ as the ...
3
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77 views

Matrix which is not similar to it's transposed

Let $V$ be vector space over a field $\mathbb{k}$. I can prove that any matrix is similar to its matrix transpose if $\mathbb{k}$ is an infinite field, but is this still true when $\Bbb k$ is finite? ...
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48 views

Matrix equation $A=PBP^{-1}$

Suppose $P$ is invertible and $A=PBP^{-1}$ . Solve for $B$ in terms of $A$. My attempt: I just left multiplied the equation by $P^{-1}$ and right multiplied it by $P$ so that I got $B=P^{-1}AP$. Is ...