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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Are points in general position generic points?

In Harris' algebraic geometry book, $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$ are said to be in general position if no $n+1$ or fewer of them are dependent. I want to prove that, if ...
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76 views

Question on the uniqueness of LU decomposition

Let $A \in \mathbb{K}^{n \times n}$ a matrix, and suppose that we can run the Gaussian elimination on $A$ without row or column interchange, so there exist the $LU$ decomposition of $A$. We define ...
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1answer
18 views

Prove the size of a linear transformation matrix

I am trying to prove that the size of a linear transformation matrix going from $R^k$ to $R^{p}$ is a $p*k$ matrix. Assuming $ p,k \geq 1$. I can prove it for fixed values of $p$ and $k$ but I am ...
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235 views

Linear Algebra : Invertible Matrix Proof

I was doing some linear algebra exercises and came across the following tough problem : Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose ...
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77 views

$T,S: V\to V$, prove that $TS$ and $ST$ have the same eigenvalues

hey I was trying to prove this proposition by dividing to cases and this is what I've got so far: let's assume without loss of generality that T is invertible: ST = [T][S] TS = [S][T] ...
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20 views

Proof of linear transformation $\mathbb{\tilde{L}}(u)=\mathbb{L}(u)$

I'm working on linear transformation and trying to answer : Let $E$ and $F$ be two vector spaces on $\mathbb{K}$, $E$ is finite. $V \ \subset E$ a subvector space of E. $L \ \in \mathbb{L}(V,F)$ ...
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189 views

Show that the additive inverse condition can be replaced by $0v = v$ for all $v \in V$

In the definition of a vector space, the additive inverse condition requires that for every $v \in V$ (where $V$ is a vector space over $\mathbf{F} = \mathbb{R}$ or $\mathbb{C}$), there exists ...
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2answers
143 views

When can the rank of a submodule be bigger than the rank of the module itself?

It is well known that the dimension of a subspace is less than or equal to the dimension of the vector space it is contained in. The same is true e.g. for modules over a principal ring. I am looking ...
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38 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 ...
2
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1answer
64 views

how to prove if [T]b is diagonal then there is a scalar “a” such that T(v)=av

hey i was trying to prove the next proposition: given T:V->V for every Basis B, if the matrix [T]B is diagonal, then there is a scalar "a" for every v in V such that T(v)=av this is what i managed ...
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1answer
46 views

Norm of integral operator in $L^1(0,2)$

How exactly do I show that an integral operator is bounded. For example, consider the following operator $$ T: L^1(0,2) \to L^1(0,2)\\ (Tf)(x):=\int_0^x tf(t) dt$$ My first approach \begin{align} ...
3
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1answer
68 views

To Show$ Ax=y$ has no Solutions given that A is a 3x3 non invertible matrix

I am trying to answer the following question: Given that A is a 3x3 matrix where the last row is the sum of the first two rows show that$ Ax=y$ has no solutions. $y \in R^3$ I was thinking that ...
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3answers
63 views

How does this reduced matrix indicate that the vectors are linearly independent?

I know that a set of vectors is linearly independent when a linear combination of them equal to zero is only satisfied by coefficients that are all zero. For this particular question, we have a ...
4
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1answer
57 views

Name of the LU decomposition algorithm

On the wikipedia page of LU decomposition there is an algorithm that produce the decomposition. It is called Doolittle algorithm. I'm really interested who is Doolittle? Or from where the name comes ...
2
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1answer
35 views

finding eigenvalues of a linear transformation and determine if its diagonalizable

Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that $T(a_1,a_2,a_3) = (-3a_3, a_1+5a_3,a_2-a_3)$. (i) Find all the eigenvalues of T (ii) For each eigenvalue ...
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1answer
39 views

Show positive definite

On the complex vector space $C([0,1], \mathbb{C})$ one defines a scalar product $<f,g>:=\int_0^1 f(t)\bar{g(t)}dt$. How do I show that this bilinear form is positive definite? I. e. how do I ...
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59 views

When can vectors of one basis be expressed as linear combination of vectors of another basis with unitary matrix coefficients?

If I have two normalized basis $\{v\}$ and $\{w\}$ for the same hilbert space of dimension $n$ ( not necessarily orthogonal ), then when can we write the following $$v_i=\sum c_{ij}w_j.....(1)$$ such ...
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1answer
41 views

Mapping of the eigenvector of eigenvalue 1 to a different matrix

Let $M \in (0,1)^{n\times n}$ be an irreducible and primitive column stochastic matrix. Then for the Perron theorem, $\exists x^* : Mx^* = x^*$. We want to build a matrix $K \in \mathbb{R}^{n\times ...
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36 views

show that some differential operator has infinitely many eigenvalues

Let $V=C(-\infty,\infty)$ be the set of all functions such that the nth derivative is continuous on $\mathbb{R}$ for all $n\geq 1$. Let $T:V\to V$ be a linear transformation such that $T(f(x)) = ...
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2answers
61 views

$A$ is normal matrix and has distinct eigenvalue, and $AB=0$. why $B$ is normal matrix?

Let $A,B \in {M_n}$ . suppose $A$ is normal matrix and has distinct eigenvalue, and $AB=0$. why $B$ is normal matrix?
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197 views

The colimit of all finite-dimensional vector spaces

Let $\mathsf{iFinVect}_K$ be the category of finite-dimensional vector spaces with injective linear maps and $X : \mathsf{iFinVect}_K \to \mathsf{Vect}_K$ be the inclusion functor. Then ...
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1answer
32 views

Given $x =(x_1,x_2,x_3)T$ and $y =(y_1,y_2,y_3)T$ determine which of the following are inner products for$ R3×1$:

Given $\bf{x}$ $= (x_1, x_2, x_3)^T$ and $\bf{y}$ $= (y_1, y_2, y_3)^T$ determine which of the following are inner products for $R^{3 \times 1}$: a) $<\bar x, \bar y> = x_1y_1 + x_3y_3$ b) ...
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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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1answer
207 views

Verify Cauchy-Schwarz holds for two vectors.

How can I show that $(\alpha_1 + \alpha_2 + \ldots + \alpha_n)^2 \le n(\alpha_1^2 + \alpha_2^2 + \ldots + \alpha_n^2) \ \text{for} \ \alpha_i \in \bf{R}$? My hint is to invoke Cauchy-Schwarz with ...
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1answer
14 views

Confusion about the number of solutions of a linear system

In the lectures, the rank of a matrix $A$ was defined to be the number of non-zero rows in the matrix obtained after reducing $A$ to row echelon form. Then the lecturer stated that if ...
3
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1answer
175 views

What is the purpose of Wronskian (linear independence/variation of parameters)?

So as I understand, the Wronskian determinant can be used to show linear independence. Why is that? Also, how does this fit into understanding variation of parameters for solving a differential ...
0
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1answer
101 views

Find the dual of the given primal linear programming problem

The primal problem is as followed: Minimize $z=4x-5y$ Subject to $y\le10-x$, $y\le2+3x$, $x,y\ge0$ Write out its dual and solve it geometrically. ...I have found its dual and graphed out the ...
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1answer
76 views

Basis of a Z-module

I think I might know how to start this problem but I'm not sure how to finish. Here is the statement: Determine a basis for the ℤ-module of integer solutions to the following system of equations: ...
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2answers
56 views

An idempotent bounded linear operator has eigenvalues $0,1$

I am thinking of the following problem: suppose $T$ is an idempotent bounded linear operator on a Banach space $X$ over the complex field. Of course, suppose $T$ is not zero map or identity map to ...
2
votes
3answers
181 views

If W is a T-invariant subspace of V and W' is a subspace of W, is W' also T-invariant?

If W is a T-invariant subspace of V, then T(W) is contained in W, but I'm not really sure where to go from here.
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68 views

Finding inverse of matrix with trig functions

$$\begin{bmatrix} \cos(30^\circ) & 0 \\ \sin(30^\circ) & 1 \end{bmatrix}$$ I am fine with finding the inverse of a standard $2 \times 2$ matrix but I am struggling to find the inverse of this ...
3
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1answer
70 views

Collection of matrices with every vector fixed

Suppose that $S$ is a collection of $n\times n$ matrices closed under addition, multiplication, and scalar multiplication, such that every vector is fixed by some element of $S$. Must the identity ...
2
votes
1answer
44 views

Working on pivots intuition

I am learning linear algebra using Gilbert Strangs "Intro to LA" 4th edition. On problem set for chapter 2.3 "Elimination Using Matrices" I encountered a question I can't wrap my head around(problem ...
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2answers
62 views

Find all vectors orthogonal to two parallel vectors

Find all vectors $\vec{v}=\begin{bmatrix}x\\y\\z\end{bmatrix}$ orthogonal to both $\vec{u_1}=\begin{bmatrix}2\\0\\-1\end{bmatrix}$ and $\vec{u_2}=\begin{bmatrix}-4\\0\\2\end{bmatrix}$ $\vec{u_1}$ ...
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2answers
26 views

How do you prove the dimension of a subspace?

How do you go about this? I just need a little nudge in the right direction. Prove that $v = \{(x,y,z): x + 3y + 5z = 0 \}$ has a dimension of 2.
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1answer
118 views

Relationship between Schur vectors and eigen vectors

If $A \in \mathbb C^{n \times n}$ has distinct eigenvalues and $AB = BA$ what can I say about the Schur vectors of $B$? I can see that $A,B$ share eigen vectors but this doesn't say anything about ...
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2answers
111 views

Find basis so matrix is in Jordan Canonical Form

$M = \left(\begin{array}{ccc}0 & -3 & -2 \\1 & 3 & 1 \\1 & 2 & 3\end{array}\right)$ I want to find a basis $B$ such that matrix for $M$ w.r.t $B$ has the form: ...
3
votes
1answer
45 views

Show that there exists α ∈ F such that S = αT.

Let $V$ be a (possibly infinite-dimensional) vector space over a field $F$. Let $S : V \to F$ and let $T : V \to F$ be linear transformations. Assume that $N(S) \supseteq N(T)$. Show that there exists ...
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2answers
46 views

Finding a vector that is orthogonal to two other vectors

Say that j = \begin{pmatrix} 2 \\ 5 \\ -1 \end{pmatrix} and k = \begin{pmatrix} -6 \\ 4 \\ -3 \end{pmatrix} There are two nonzero 3D vectors a = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, which are ...
3
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2answers
520 views

Eigenvalues and Spectrum

In algebra, I learned that if $\lambda$ is an eigenvalue of a linear operator $T$, I can have \begin{equation} Tx = \lambda x \tag{1} \end{equation} for some $x\neq 0$, which is equivalent to $\lambda ...
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1answer
28 views

$\Bbb K$-algebra structure on isomorphic vector space

Let $V,W$ be two isomorphic vector spaces over a field $\Bbb K$. Suppose we find an operation $\star_V$ which makes $V$ a $\Bbb K$-algebra. Then I think even $W$ get the $\Bbb K$-algebra structure, ...
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125 views

What is the difference between a vector and its transpose?

I have seen a definition of orthogonality between two vectors something like this: $$<\vec{u}, \vec{v}> = u^T \cdot v = 0$$ I am wondering what is the purpose of using a transpose of a vector ...
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2answers
136 views

How do I prove that $\{AB-BA\}$ does not span matrix space?

Let $F$ be a field. Let $W$ be the subspace of $M_n(F)$ generated by elements of the form $AB-BA$. How do I prove that $dim(W)<n^2$?
3
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1answer
102 views

Is the norm operator between normed spaces ever induced from an inner product?

Assume $(V,\| \|_V),(W,\| \|_W)$ are both finite dimensional normed spaces. We have the induced operator norm on $Hom(V,W)$. When does it occur that this norm is actually induced from some inner ...
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2answers
107 views

Are the geometric sequences a subspace of all infinite sequences of real numbers

I'm a little confused about how to prove that the given infinite sequence is a subset of "the space of all infinite sequences." so V = subspace of all infinite sequence of real numbers so $$(x_0, ...
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2answers
82 views

Why does $\mathrm{Rank}(A{A^*} - {A^*}A) \ne 1$?

Given $A \in M_n$, why does $\mathrm{Rank}(A{A^*} - {A^*}A) \neq 1$?
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1answer
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How do I solve this matrices problem?

Well,I need to find for which values of the parameter a the the rank of the matrices is :- rank=1 rank=2 rank=3 rank=4 note that the parameter a is a complex number. I tried to solve it but all ...
12
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2answers
1k views

If $A^2$ and $B^2$ are similar matrices, do $A$ and $B$ have to be similar?

I know that the converse is true; that is, if A and B are similar matrices, then $A^2$ and $B^2$ are similar . However, I'm not sure about the reverse.
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3answers
57 views

What is a counterexample to this one?

Let $R$ be a commutative ring and $A\in M_{n\times m}(R)$ where $n\neq m$. What is an example such that $\det(AA^t)\neq \det(A^tA)$? Indeed, I think it's true. If this is true, how do I prove this?
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41 views

Help on symmetric matrices question

Hello I am having trouble coming up with the solution to a problem from the book "Schaums Outlines: Linear algebra" The answer is in there, but not the solution The question is, find a real ...