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 ...

learn more… | top users | synonyms (1)

2
votes
1answer
25 views

Linear Algebra: orientations of vector spaces (problem)

This is an exercise from J.Munkres's Analysis on Manifolds: Consider the vectors $\mathbf{a_i}$ in $\mathbb{R}^3$ such that:$$[\mathbf{a_1},\mathbf{a_2},\mathbf{a_3},\mathbf{a_4}]=\begin{bmatrix} 1 &...
1
vote
1answer
36 views

Matrix Decomposition Question: If $A$ is symmetric when is it true that $A = B^T B $

I was thinking about the fact that for a symmetric matrix A the maximum values of $X^T A X$, where $\|X\| = 1$ occurs when $X$ is an eigenvalue of $A$. I was trying to think of a good geometric image ...
1
vote
2answers
52 views

Undergraduate Linear Algebra Problem

Prove that if $A$ and $B$ are matrices such that $I - AB$ is invertible, then the inverse of $I - BA$ is given by the formula $(I - BA)^{-1} = I + B(I - AB)^{-1}A$ People, please help!
1
vote
3answers
281 views

BMO2 Question 1 2015 Iterative Formula Problem

$1.$ The first term $x_{1}$ of a sequence is $2014$. Each subsequent term of the sequence is defined in terms of the previous term. The iterative formula is $x_{n+1} = \frac{(\sqrt2 + 1)x_{n} − 1} {(\...
-2
votes
2answers
163 views

$A^2 B=A $ iff $B^2 A=B$ [closed]

Given that $ A, B$ are complex square matrices of size $n$ having same rank, then $A^2 B=A $ iff $B^2 A=B$. Thanks for helping, I am totally out of clue about the problem.
2
votes
1answer
113 views

Affine subspace of $\Bbb{R}^{27}$

I have affine subspace $K$, $K \subset \mathbb R^{27}$. It's elements are solutions of system of linear equations $Ax=b, b \in R^{16}$. What are maximum and minimum dimensions of said subspace, if I ...
6
votes
2answers
77 views

Order $n^2$ different reals, such that they form a $\mathbb{R^n}$ basis

I've been trying to solve this linear algebra problem: You are given $n^2 > 1$ pairwise different real numbers. Show that it's always possible to construct with them a basis for $\mathbb{R^n}$. ...
0
votes
2answers
137 views

Has the point dimension zero or one?

My question is really simple. If $K$ is a field, We know that the subspaces of $K^{n}$ generated by the point $(x_1,\ldots,x_n)\neq (0\ldots,0)$ has dimension one. However, if the generator point is $(...
2
votes
0answers
44 views

Eigenvalue and informations?

I have been wondering about this these last couple of days and cannot seem to understand how come this is the case.. I know about the eigevalues and the eigenvector, and that it can be found on for ...
1
vote
2answers
79 views

Find image under $T$ of the line $x_1 + 2x_2 = 3$ — Linear Algebra

We are asked to find the image under $T$ of the line $$x_1 + 2x_2 = 3$$. Consider the linear operator $T:\mathbb{R^2}\rightarrow \mathbb{R^2}$ with standard matrix $$ A = \left[\begin{array}{rrr} ...
1
vote
3answers
470 views

How do I determine whether a set spans or does not span a vector space?

$\{1+x^2,1+x+2x^2,x+x^2\}$ in $P_2$ Does the set span $P_2$? I understand that the set is linearly dependent but how can I prove that the two independent elements cannot form every element in $P_2$? ...
3
votes
1answer
52 views

If $\| v \| = \| T(v) \|$ for all $v \in V$, then $T$ is onto.

Assume that $V$ is finite dimensional. $T: V \rightarrow V$ is linear. I know that if $\| v \| = \| T(v) \|$ for all $v \in V$ then $T$ is unitary. That is, $\langle T(x),T(y)\rangle=\langle x,y\...
2
votes
0answers
30 views

Linear Transformations and Composition

I have this question: 1 - Let V be a vector space and T a linear operator $T:V\rightarrow V $, show that $$[T^m]_B =[T]_B^m$$ Where $B$ is a basis(any) of $V$ and $T^m=T\circ T \circ T\circ....\circ ...
0
votes
1answer
65 views

Linear Map is Homeomorphic in $\mathbb R^k$

I am having trouble understanding a proof in Rudin's "Real and Complex Analysis." The theorem states that To every linear transformation $T$ of $\mathbb R^k$ into $\mathbb R^k$ corresponds a real ...
3
votes
4answers
89 views

Show that $A$ and $A^T$ do not have the same eigenvectors in general

I understood that $A$ and $A^T$ have the same eigenvalues, since $$\det(A - \lambda I)= \det(A^T - \lambda I) = \det(A - \lambda I)^T$$ The problem is to show that $A$ and $A^T$ do not have the same ...
1
vote
1answer
48 views

Splitting condition for iteration method

In iteration methods to solve $Ax = b$ we have the standard form as $$M x^{(k+1)} = N x^{(k)} + b \tag{$*$}$$ which $A = M - N$, and $M^{-1}N$ is called iteration matrix. The standard convergence ...
0
votes
2answers
43 views

find example of linear transformation

Q: let $n \geqslant 3$, find an example of linear transformation $T: R^n \to R^n$ that $Im \cap ker T =\{0\}$ and that $\text{ker}\ T=\{(x_1,x_2,...,x_n) | x_1+x_2+...+x_{n-2}=0 \}$. I'm pretty sure ...
1
vote
1answer
50 views

If $\vec{v}$ is an eigenvector of $A$, then also $B\vec{v}$, when $AB = BA$ [duplicate]

I have the following problem: Let $V$ be a finite dimensional vector space. Let $A$, $B$ be linear maps of $V$ into itself. Assume that $AB = BA$. Show that if $\vec{v}$ is an eigenvector of $A$, ...
5
votes
1answer
64 views

Kernel of a matrix pencil

Let $A,B$ be $n\times n$ singular real matrices such $ker A\cap ker B=\{0\}$, how could I show that there exists $x\in \mathbb R$ such that $ker (A+xB)=\{0\}$?
1
vote
1answer
38 views

Determining span, is there an easier way to remember it?

As I understand it, to set up a problem to determine if the vector spans $ \mathbb{R}^n$ or if the given vector is in the Span of $$(v_1,v_2,...,v_n)$$ you take the vector and set up an augmented ...
2
votes
3answers
36 views

quadratic equation plot investigation

Let $f(x) =-x^2-4x+18 $ so i plot it like this : But my imagination created the following: $-x^2-4x+18=0 -> x^2+4x = 18-> x^2+4x-18 = 0$ Which yields the parabola upside down. Where's the ...
1
vote
2answers
116 views

Minimum distance between Vector and its projection

I'm struggling to get to an easy and simple algebraic solution for this question: I actually thought about optimization with the Cauchy-Schwarz Inequality but it just got dirty and probably wrong. ...
1
vote
1answer
58 views

Orientations of $n$-dimensional vector spaces

Let $V$ be a $n$-dimensional vector space. An $n$-tuple $(\mathbf{a_1},...,\mathbf{a_n})$ of independent vectors in $V$ is called an $n$-frame in $V$. In $\mathbb{R}^n$ we call such a frame right-...
2
votes
0answers
26 views

Maximum number of MUBs is less than or equal to $n+1$

We define mutually unbiased bases (MUBs) as Let $\mathcal{B}_1, ..., \mathcal{B}_k$ be $k$ orthonormal bases for $\mathbb{C}^n$. We call these bases mutually unbiased if they obey $$ \forall v \...
3
votes
2answers
47 views

Finding the maximum value of $\frac{5}{4x^{2}-16x+21}$

Determine the maximum value of $$\frac{5}{4x^{2}-16x+21}$$. I tried completing the square to get $$\frac{5}{4(x-2)^{2}+5}$$ But I'm struggling to proceed. Any hints? NO CALCULUS PLEASE.
3
votes
1answer
50 views

Let $V$ be an inner product space. Show that if $||x+y||=||x||+||y||$, then $ax=by$ where $a,b$ are non-negative and not both zero.

Let $V$ be an inner product space. Show that if $||x+y||=||x||+||y||$, then $ax=by$ where $a,b$ are non-negative and not both zero. I know that the converse is true. I considered the square of the ...
0
votes
0answers
24 views

A matching M in a graph G is a maximum matching if and only if G has no M -augmented path.

Hi there is a theorem like this and there is a proof of that theorem. But i did not understand the proof even if i google it. Can anyone can help me with an understandable explanation? Theorem: A ...
3
votes
1answer
81 views

Find all possible Jordan Canonical forms of $A^2$

Finding Jordan Canonical forms seems pretty straightforward mostly, but this one threw me off: Let $A\in M_n(\Bbb{F}) $ be a matrix with a minimal polynomial $m_A(t)=(t-\lambda)^n$. Find all the ...
2
votes
0answers
22 views

Determining if a function is a linear transformation

$$T(p(x)) = p(x+1)$$ $$T: R[x] \to R[x]$$ Let's take $p(x), q(x) \in R[x]$ Now I need to check if $T(p(x)+q(x)) = T(p(x)) + T(q(x))$ From the definition I know that $T(p(x)) + T(q(x)) = p(x+1) + q(...
5
votes
2answers
518 views

Proof that determinant is continuous using $\epsilon - \delta $ definition

I need to prove that the determinant $\det: M(n, \mathbb{R}) \rightarrow \mathbb{R}$ is a continuous function given the euclidean metric on the vector space of all $n x n$ matrices over $\mathbb{R}$, ...
1
vote
2answers
55 views

question in linear algebra on Hermitian matrices

Hello this indeed a very short question from Algebra that I have no real idea on and figured it is simple but for some reason I cannot seem to find it. I am given $A$ and $B$ complex square matrices ...
5
votes
3answers
175 views

Inverse of a matrix having zeroes in diagonal and one elsewhere

Could any one help me to find inverse of such matrix? I observed that $A= J-I$, where J is a matrix having all entries 1. Thanks for helping.
1
vote
4answers
70 views

Rank of $A^2$ is $A$ is symmetric

Let $A$ be a symmetric matrix. Is $rank(A^2)=rank(A)$ I'm pretty sure this is true, and if it is, could someone guide me towards the method of proving it?
0
votes
5answers
225 views

$AB-BA=A$, Then A is singular? [duplicate]

Title is the question, I tried taking trace both side and got trace of $A$ is zero, now to conclude $A$ is singular, suppose $A$ is non singular, then multiplying both side by Inverse of $A$ we get $B$...
0
votes
3answers
2k views

Trace of AB = Trace of BA

We can define trace if $A =\sum_{i} \langle e_i, Ae_i\rangle$ where $e_i$'s are standard column vectors, and $\langle x, y\rangle =x^t y$ for suitable column vectors $x, y$. With this set up, I want ...
-2
votes
1answer
109 views

how to prove that a composite function is a linear transformation? [closed]

I was studing linear algebra, being more specific linear transformation, I think that a composite function is a linear transformation, but how to prove it?
2
votes
0answers
24 views

What is an intersection matrix and how it works?

At this page there is a doc about how to handle spatial coordinates and shapes, the figure shows an intersection matrix and according to the writer this matrix generates a bit pattern. \begin{matrix} ...
2
votes
1answer
100 views

How to turn this matrix to Jordan normal form?

Matrix $A$ is $ \left( \begin{array}{ccc} 3 & 0 & 8 \\ 3 & -1 & 6 \\ -2 & 0 & -5 \end{array} \right)$ and I need to find a matrix P such that $P^{-1} A P = J$ where $J$ is a ...
3
votes
1answer
58 views

Difficulties understanding these statements about change of basis

I understood more or less what a change of basis matrix is and how I can use it to pass to one coordinate system to another. Basically, a change of basis matrix is a matrix whose columns are the ...
1
vote
2answers
55 views

Are all $k$-vectors in $\mathbb{R}^3$ simple?

The second comment of the accepted answer here, claims that all $k$-vectors in $\mathbb{R}^3$ are simple, that is, they can be written as the wedge product of k vectors. I tried to show this for $2$-...
0
votes
1answer
31 views

Let the system $Ax=b$ be incompatible. Prove that $C^kx=0, C=[A,b]$ is determined for all $k\in \Bbb{N}$.

Let $A \in \Bbb{R}^{n \times (n-1) }$ be of rank $n-1$, let $b\in \Bbb{R}^n$. Let the system $Ax=b$ be incompatible. Prove that $C^kx=0, C=[A,b]$ is determined for all $k\in \Bbb{N}$. I can't use ...
1
vote
1answer
38 views

inner product space problem $(x_n,y_n)\to 0$

If $(y_n)$ is a bounded sequence in an inner product space, and $(x_n)$ is a sequence converging to zero, prove that $(x_n,y_n)\to 0$. Where $(x_n,y_n)$ is the inner product. Since $(y_n)$ is bounded ...
1
vote
0answers
46 views

Extensions of Vector Spaces

I have just finished learning about field extensions and such. In particular I am interested in the minimum polynomial of $a$ in some field extension $K$ of $F$. Particularly I have learned about the ...
0
votes
3answers
51 views

Is this space complete or is it incomplete?

Show (if possible) that the space of all complex sequences $x=(x_n)$ with only a finite number of terms nonzero (the number of nonzero terms may be different for different members of the space) is ...
0
votes
1answer
54 views

Magnitude of product of symmetric matrix and unit vector

If $A$ is any symmetric 2 by 2 matrix with eigenvalues -3 and 3 and $\vec{u}$ is a unit vector in $\mathbb{R}^2$, what is $||A\vec{u}||$? Any help would be appreciated, I haven't the slightest idea ...
1
vote
1answer
65 views

How to find $r$ in an equation like this: $r^3= xr+y$

Can anyone give me an an idea how to solve this and find $r$, where $r^3= xr+y$ and $x$ and $y$ are known numbers?
1
vote
1answer
34 views

Is there such a thing as a weighted multiple regression?

I'm new to linear algebra, but I know how multiple linear regressions work. What I want to do is something slightly different. As an example, let's say that I have a list of nutrients I want to get ...
0
votes
1answer
28 views

Linear Independence of a given set

The Group ${xcosx,x,cosx}$ is linear independent i cant understand why? I did managed to prove it was linear dependent by saying that $xcosx=λcosx$ <=> $λ=x$ How can I prove that this group is ...
1
vote
1answer
55 views

Vector Subspaces Proof

I'm having a hard time trying to prove the following question: I tried to use de definition of Span, but it just got redundant and got me running in circles.. Here`s what I've got so far... ...