Tagged Questions

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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Equation for minimum/maximum eigenvalue

It is well known that for a hermitian matrix $A$ we have $\lambda_{min}(A)=min{x\ne 0} <x,Ax>/<x,x>$, which we can see be diagonalizing $A$. Now here is my question about the following I ...
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“The limit of a sequence is insensitive to finite changes in the sequence” - help me understand this sentence!

The context is p59 of "A primer on Hilbert Space Theory, by Alabiso and Weiss, where the example is given of a quotient $c_{00}/c_0$ which is "the quotient of the space $c_0$, the space of all ...
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Prove that $tr(A^-)=\sum_{i=1}^n\lambda_i^{-1}$

If $A$ is an n$x$n symmetric matrix with $r$ nonzero characteristic roots $\lambda_1,\lambda_2,...,\lambda_n$ and $A^-$generalized inverse of $A$ (not $A^{-1}$), then ...
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Typo in the book or am I going crazy?

I am reading about integral bases from Frazer Jarvis' "Algebraic Number Theory", but my question is really about elementary linear algebra. In page 49, author claims the following: I don't think ...
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What is wrong with this parametrization?

I need to find the $N$-by-$1$ vector $\mathbf{x}$ that minimizes the following expression: $L=\alpha |\hat{\mathbf{H}}_{1}\mathbf{x}|^2 +(1-\alpha)|\mathbf{H}_{2}\mathbf{x}-\hat{\mathbf{Y}}_{2}|^2$, ...
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25 views

Does Linearity imply Commutativity?

If I have two linear operation X and Y , could I conclude that for X + Y = Y + X ?
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How to prove the operator D=d^(4)/dx is self adjoint

I'm trying to prove $D=d^{4}/dx$ is self adjoint, I think it is trivial but the book let me use Lagrange identity to show it.
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1answer
12 views

Minimal polynomial in $T$-invariant subspace

I am stuck on the following problem. Problem: Let $V$ be a finite dimensional vector space over field $F$ and $T$ a linear transformation from $V$ to $V$. $W$ is an invariant subspace. Let $h_1$ be ...
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1answer
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Rational Canonical Form Confusion; Choosing Basis Which Gives the Rational Canonical Form.

I am reading the theory of finitely generated modules over a PID. One of the applications of the the theory is that one can derive the theory of rational canonical form of a linear operator on a ...
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2answers
55 views

$\det (AB)=\det(A)\det(B)$ is possible when $A$ and $B$ are _____?

$\det (AB)=\det(A)\det(B)$ is possible when $A$ and $B$ are _____? This is a fill-in-the-blank problem that I found in my paper, but I don't have this answer.
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Non-orthogonal basis

I have a set of complex vectors (maybe 10,000 vectors, each of which has maybe 200 elements). I know that each of the complex vectors is a linear combination of a small (maybe 10) collection of ...
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21 views

Orthogonal Matrix with a specific row

I have an assignment with the following question: Does an Orthogonal Matrix exist such that its first row consists of the following values: ($1$/$\sqrt{3}$, ...
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3answers
26 views

Help understanding a proof about vector spaces

The exercise goes like this: -Let $W= {(x,y,z)|2x+3y-z=0}$ Then $W\subseteq\mathbb{R}^3$, find the dimension of $W$. -Find the dimension $[\mathbb{R}^3|W]$ This was a problem from my algebra exam, ...
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Find the eigenvalues and eigenvectors of T in V

Let $\mathbf{V}$ be the linear span of the functions 1, cos x, sin x. Let the operator T on V be given by the rule $T y(x)= y(x+\pi/4)$. Find the eigenvalues and eigenvectors of T in V. I'm not sure ...
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1answer
19 views

What is the purpose of Jordan Canonical Form?

I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix. What is ...
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How to put a matrix in Jordan canonical form, when it has a multiple eigenvalue?

I have a question that reads: Put the matrix \begin{bmatrix} 3 & -4\\ 1 & -1 \end{bmatrix} in Jordan Canonical Form. Moreover, in each case, find the appropriate ...
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Covariance Matrix Proof - Confusion with Cov(X,X) = Covmat(X)?

I have completed a proof regarding variance, covariance, and the covariance matrix. I think I have made a mistake regarding an assumtion. I need to show that $var(\{f\}) = a^TCovmat(\{x\})a$ Where ...
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1answer
21 views

Finding a function for othorgonality

Let the polynomial $f$ of the form $f : t \rightarrow a_0 + a_1t + a_2t^2 $. Find the function $f$ such that $f$ is orthogonal to $t$, $t^2$ and $\|f\| = 1$. I got stuck with finding function. Here ...
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2answers
38 views

Is following set closed under multiplication? [on hold]

Is the set $\{-2,0,2\}$ closed under multiplication. If not, justify your answer with a counterexample.
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1answer
24 views

Isomorphism between $\operatorname{O}(A)$ and $\operatorname{SO}(A\times \{0\})$ for $A \subset \mathbb{R}^2$

I was given this exercise and to be honest I can't wrap my head around this one at all. Maybe some of you can shed some light on the problem at hand. I don't want a full solution, but some hints would ...
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4answers
33 views

Find eigenvalues and eigenvectors of the operator $A$

The question is: Find the eigenvalues and eigenvectors of the operator $A$ on $\Bbb{R}^3$ given by $A\mathbf{x}=|\mathbf{a}|^2 \mathbf{x}- (\mathbf{a} \cdot \mathbf{x}) \mathbf{a}$, where $\mathbf{a}$ ...
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3answers
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Determining if a set is in the subspace of a continuous function

Let $A={\rm span}\{\cos^2x,\sin^2x\}$ be a subspace of the set of functions $C[0,\pi]$, for each of the following functions in $C[0,\pi]$, determine whether or not it is in $A$. $f(x)=1$ ...
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2answers
52 views

Prove $W \cap W^\perp =\{\vec{0}\}$

If $W$ is a subspace of $\mathbb{R}^n$, then $W^\perp = \overline{W} = \{v \cdot w = 0, \forall w \in W\}$ Prove $W \cap W^\perp = \{\vec{0}\}$. How do I fully prove this intersection is ...
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3answers
60 views

A is an $n \times n$ matrix such that $A^2 = A$

I was doing the final homework of the term and got to the last question thinking I was gonna cross the finish line with ease until I got to the last two questions. The second last question, I ...
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24 views

Find Eigenvalues of Infinite Matrix

I have the matrix $M$ acting on $l^2(\mathbb{N;C})$ given by the components $$ M_{n,n'} = V_n\delta_{n,n'} + A\delta_{n,n'+1}+A^\ast\delta_{n,n'-1} $$ where $V_n$ is real and obeys a periodicity ...
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2answers
27 views

$rank(T^n) = rank(T^m)$ for any positive integer $m \geq n$

Let $T$ be a linear operator on a finite dimensional space $V$. Prove that if $rank(T^n) = rank(T^{n+1})$ for some positive integer $n$, then $rank(T^n) = rank(T^m)$ for all positive integer $m \geq ...
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0answers
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What's a good primer from linear algebra to spherical harmonics?

I need a topic, a primer, that will be able to introduce me to spherical harmonics and how to translate and use them with the usual tools of linear algebra and calculus, namely matrices, polynomials ...
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1answer
29 views

How do I algebraically express that I'm using the integer part of a real number?

I need to elaborate an answer and I need to display that from real number (e.g. 1.236) the answer would be just the integer part (e.g. 1), how would I do that, there is something like $abs()$, but for ...
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1answer
20 views

Inverse matrix as a sum of matrix powers [duplicate]

I have matrix $ A\in \mathbb{C}^{n x n}$ and $A$ is invertible. How can I show that coefficients $c_0,...,c_{n-1}$ exist : $A^{-1} = c_0I+c_1A+...+c_{n-1}A^{n-1}$ I tried to solve it first by ...
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1answer
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Eigenvalues of Sub-Matrix Formed from subset of Columns

I have an n-by-p matrix $X$ and I consider the eigenvalues of the p-by-p matrix $X^{'}X$. Let's denote the largest and smallest eigenvalues of $X^{'}X$ with the usual notation $\lambda_{1}(X^{'}X)$ ...
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2answers
19 views

How to determine the smallest interpolation degree required?

Given a set of $n$ points $(x_k, y_k)\ (k\in\{1,...,n\})$, of course a polynomial of degree $n$ can fit all points. However, in some cases the coefficient of the higher degrees actually vanish and one ...
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0answers
24 views

Orthonormal basis for a subspace of P2

I was wondering if somebody could help please. I have the following question: Let $f(x) = -2, g(x) = -8x+6, h(x) = -9x^2-4x+5$ and consider the inner product $< p(x), q(x) > = p(-1)q(-1) ...
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0answers
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Vector space or no? [on hold]

For the set of positive real numbers, addition and scalar multiplication will be as follows: $$y+z = yz$$ $$ky = y^k $$ Is this a vector space?
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1answer
18 views

Need help regarding Subspace of matrix and its basis

I need some kind of hint to get me going for this question as I'm so lost at it. Any sort of help would be appreciated. Let E be the set of all 2x2 matrices that have $v={(1,-1)}$ as an eigenvector. ...
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Find solution of signle element $y_i$ in vector $y$ subject to $Ay=c$

I have a interesting question about linear algebra problem. Assume that I have a matrix $A^{m \times n}$ and vector $c^{n \times 1}$ are known and I want to find the solution of vector y subject to ...
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1answer
25 views

Induced invariant linear map in the dual space

This is the problem that I am stuck on. Problem: Let $V$ be a finite dimensional vector space and $T: V\rightarrow V$ be a linear transformation. Suppose ...
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0answers
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how that T is ergodic if and only if the only eigenfunctions $f \in L^2(\mu)$ of $U_T$ corresponding to the eigenvalue $1$ are constant functions.

Let $T:X \rightarrow X$ be a measure-preserving transformation. Assume that $(X,\mathcal{B},\mu)$ is a probability space. Show that T is ergodic if and only if the only eigenfunctions $f \in ...
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5answers
100 views

If $A^n=0$, then $I_n-A$ is invertible. [on hold]

How do I solve this problem? $A$ is $n\times n$ and $A^n=0$. Prove that $I_n-A$ is invertible.
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1answer
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column vector dot product with transpose

Say I have an orthonormal base, B = {v1, v2, ... vn} for space Rn. Assuming v1 is a row vector, what is $vi^t \cdot vi$ ? Is it a scalar, or is it an nxn matrix with a 1 in the [ ]$_{i,i}$ spot?
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1answer
22 views

Resolve $Ay=b$ with fast method

I am looking for method that resolve equation $$Ay=b$$ I read the paper "Wiedemann's algorithm" that is one solution for fast way to find the solution instead of Gauss-Elimination. Could you suggest ...
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1answer
37 views

Is this a basis for 2x2 matrices?

I am completely lost. I tried getting help, but this doesn't make sense. $2 \times 2$ matrix: $$ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} $$ How do I prove a basis for this? I've ...
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1answer
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Subset generated for subset.

Let $S\subset \mathbb R^n$. Suppose that $S$ is not contained in any proper subspace of $\mathbb R^n$. Thus $\mathbb R S=\mathbb R^n$ and we may select a basis $v_1,\ldots,v_n$ of $\mathbb R^n$ ...
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2answers
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Liner Algebra - Stationary Sequence

Let $T\in L(\Bbb{V},\Bbb{V})$ with $\Bbb{V}$ a $\Bbb{K}$-vector space and $\dim_{\Bbb{K}}\Bbb{V}=n<\infty$. Proof that the following sequence $$\ker T\subset\ker T^2\subset \dots \subset \ker T^l ...
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Systems of First Order Linear Equations - Differential Equations

Consider the vectors $x^{(1)}(t) = (t,1)$ and $x^{(2)}(t) = (t^2, 2t)$ I computed the Wronskian which is t^2. But I was wondering how to solve the following questions: 1) In what intervals are ...
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1answer
14 views

Vector space generated

Let $(L,+)<(\mathbb R^n,+)$ be a additive subgroup and let $\{v_1,\ldots,v_m\}$ be a maximal linearly independent subset of $L$. Let $V$ be the subspace spanned by $\{v_1,\ldots,v_m\}$. Asumme that ...
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0answers
21 views

An analytic characterization of eigenvalues of a Hermitan matrix.

If $A$ is a Hermitian matrix with eigenvalues, eigenvectors $\{ \lambda_i , e_i \}_{i=1}^n$. (where we have $\lambda_i > \lambda_{i+1}$) for a vector $v$ let its component along the corresponding ...
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16 views

Check if set of functions is a basis of space

Let $f_a \in R^R$ be function given by $f_a(x)=1$ if $x=a$ and $f_a(x)=0 $ if $x \neq a$ for $a \in R$ Decide if set of functions $f_a$ is a basis of space of functions $R^R$ ? I think I know how to ...
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Orthonormal basis and representation of a vector with angle

Given two orthonormal vectors $u_1$ and $u_2$ which form a basis for $\mathbb{R}^2$. The vector $u \in \mathbb{R}^2$ can be represent as $$u = \cos \theta \ u_1 + \sin \theta \ u_2$$ where $\theta$ is ...
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Rational Canonical Form Question

If I am looking for the Rational Canonical Form over the Real field for a matrix but have complex eigenvalues, how would I go about doing so? Any help is appreciated, thanks! Also, its for a 3 x 3 ...