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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When is “$\Re(\lambda) \gt 0$ for $\lambda \in \sigma(A),A \in \mathbb{R}^n $” true?

Let $A \in \mathbb{R}^{n \times n}$ and $\sigma(A)$ the spectrum of $A$. I am searching for a fast way to check whether $\Re(\lambda) \gt 0$ for all $\lambda \in A$. If $A = A^t$, one only has to ...
3
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2answers
751 views

Minimal polynomials and cyclic subspaces

I'm trying to make my way through two problems in Curtis's Linear Algebra, chapter 25. One of the two problems is this one, #5: Prove that $V$ is cyclic relative to a linear transformation $T \in ...
2
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1answer
40 views

If $M:=i\mathbf{\sigma}\cdot \mathbf{v}$, how do I see that $SMS^{-1}=i\mathbf{\sigma}\cdot \mathbf{v}'$

Let $\mathbf{\sigma}=\sigma_1+\sigma_2+\sigma_3$, where the $\sigma_i$ are the Pauli matrices and define: $$M:=i\mathbf{\sigma}\cdot \mathbf{v}$$ The claim is that if I change $M$ through a ...
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1answer
41 views

Algebra: Help with these expressions about inverse matrix

$x'$ meaning transpose of vector $x$. Let's say I have this expression: $$(x\cdot v')^2$$ I can write it as: $$ (x\cdot v')^2 = (x \cdot v')(x \cdot v') $$ My question came up when I saw somewhere ...
2
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1answer
58 views

Finding the Jordan basis of a linear map

A linear map $A$ is given in the canonical basis with the matrix $$ \begin{bmatrix} -2&0&-2&-2\\ 1&0&1&1\\ -1&1&-1&-1\\ 3&-1&3&3\\ \end{bmatrix} $$ ...
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1answer
39 views

Let $T:V\to W$ be linear, show $\ker T$ is a subspace of $V$ and $\operatorname{im} T = T(V)$ is a subspace of $W$

OK, so I have already proven that $\ker T$ is a subspace of $V$, which is pretty obvious because the kernel is just the $0$'s, though I'm not sure I did it formally enough. The second part I don't ...
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1answer
44 views

show there exist non zero vector which is linear combination of other

sLet $a_1, \ldots , a_n$ be a basis of linear space $V$ let $W \le V$ be a $k$ dimensional subspace $k \ge 1$ Show for each subset $\displaystyle a_{i_i}, \ldots a_{i_m}$ for $m>n-k$ exist non ...
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1answer
47 views

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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3answers
42 views

“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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47 views

Prove that $\text{span}\left\{\underline{w}_1,…,\underline{w}_n,\underline{u}\right\}=\text{span}\left\{\underline{w}_1,…,\underline{w}_n\right\}$

Suppose that $\underline{u},\underline{w}_1,...,\underline{w}_n\in\mathbb{R^n}$ $\text{span}\left\{\underline{w}_1,...,\underline{w}_n,\underline{u}\right\}=\text{span}\left\{\underline{w}_1,...,\...
4
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1answer
187 views

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 $...
2
votes
0answers
64 views

What is wrong with this parametrization?

I need to find the $N$-by-$1$ vector $\mathbf{x}$ that minimizes the following expression: $L=\alpha |\hat{\mathbf{H}}_{1}\mathbf{x}|^2 +(1-\alpha)|\mathbf{H}_{2}\mathbf{x}-\hat{\mathbf{Y}}_{2}|^2$, ...
0
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1answer
51 views

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
91 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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2answers
155 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: ($1$/$\sqrt{3}$, $1$...
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2answers
49 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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2answers
905 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 ...
2
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2answers
322 views

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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1answer
25 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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1answer
55 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
127 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
51 views

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 $A$. $f(x)=1$ $g(x)=3+\...
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2answers
383 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 $\vec{...
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3answers
73 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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2answers
48 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 $m \geq n$...
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2answers
227 views

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
36 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
102 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
104 views

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
34 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 ...
0
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1answer
41 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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0answers
36 views

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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5answers
311 views

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

How do I solve this problem? $A$ is $n\times n$ and $A^n=0$. Prove that $I_n-A$ is invertible.
0
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1answer
27 views

column vector dot product with transpose

Say I have an orthonormal base, $B = \{v_1, v_2, \ldots v_n\}$ for space $\mathbb R^n$. Assuming $v_1$ is a row vector, what is $v_i^t \cdot v_i$ ? Is it a scalar, or is it an $n\times n$ matrix ...
0
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1answer
35 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 ...
0
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1answer
78 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 ...
0
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1answer
24 views

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
23 views

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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57 views

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 $x^{(...
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1answer
31 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
140 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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1answer
386 views

If det(A) is zero, what is det(adj(A))?

I wanted to prove that det(adj(A))=det(A)^n-1 for an nxn matrix A. I separate the proof in two cases: singular and non-singular matrix A. For the non-invertible A, det(A)=0. In my head, I know that ...
5
votes
2answers
152 views

Solution of system of three variables

On solving $$ 2x - 4y + z = 0 $$ $$ x + y - 4z = 0 $$ $$ x - y - z = 0 $$ I get $$ y = 0.6 x $$ $$ z = 0.4 x$$ I thought that there was a rule of thumb that you need as many independent ...
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3answers
56 views

Evaluating a determinant for eigenvalues

I need to evaluate $$\left| {\matrix{ {3 - \lambda } & 1 & 1 \cr 2 & {4 - \lambda } & 2 \cr 1 & 1 & {3 - \lambda } \cr } } \right|$$ A direct computation ...
2
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1answer
552 views

How come that HSL can contain more information than RGB?

I have noticed weird thing when working with HSL - unlike RGB, it has some blind spots where certain value just does not matter. I'm sure we were taught about this when I had Linear algebra lectures ...
2
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0answers
76 views

Generators of group of “unitary” matrices over a finite field

This is about a group related to $U(n,q)$ and $SU(n,q)$. I know from multiple sources the generators for these groups, but $U(n,q)$ is defined to be the group of matrices $A$ such that $A^*JA = J$ ...
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2answers
20 views

can anyone explain this statement on linear dependency

When i was reading i encountered the statement"In a linearly dependent set of functions none of the function is zero function".but I cannot understand why the statement is true
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1answer
43 views

A question about the properties of the pseudospectrum

Assume that $A\in \mathbb{C}^{n\times n}$. The $\epsilon-$pseudospectrum of $A$ is defined by $$\sigma_{\epsilon}(A)=\{z\in C \quad | \quad \Arrowvert (zI-A)^{-1} \Arrowvert>\frac{1}{\epsilon}\}.$$ ...