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 ...

learn more… | top users | synonyms

2
votes
1answer
19 views

How do I find the characteristic polynomial and eigenvalues?

For the following matrix, compute its characteristic polynomial its eigenvalues $$A = \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4\end{bmatrix}$$ So I think I ...
0
votes
1answer
14 views

$r(S+T)\le r(S) + r(T)$?

If $V$ is a finite-dimensional vector space, and $S$ and $T$ are linear transformations from $V$ to $V$, how can you show that $\text{im}(S+T)$ is a subset of $\text{im}(S) + \text{im}(T)$ and also ...
0
votes
0answers
4 views

Matrix coordinate blocks and null space of a vector

Suppose we have some non-zero $c \in \mathbb{R}^n$. Let $\{U_i\}_{i=1}^m$ be a collection of $n \times n_i$ matrices where each column is a standard basis vector. Suppose that each standard basis ...
0
votes
1answer
32 views

$A^{T}b$ inconsistent system!

I am trying to figure out how the calculation on the last image comes to be (question 9, the yellow area). I have calculated the rest without issue. I know that the formula for the last set is ...
0
votes
0answers
6 views

Determine the expression for a continuous affine transformation

In this problem I'm doing, I'm being asked To determine the affine transformation matrix which maps triangle V to triangle W. I'm also being asked to determine this matrix's continuous ...
0
votes
2answers
13 views

If a composition of linear transformation is invertible, then are each linear transformations invertible?

If $S$ and $T$ are linear transformations from set $V$ to $V$, which is a finite-dimensional vector space, and if the composition $ST$ is invertible, how can we show that $T$ is one-to-one, therefore, ...
0
votes
0answers
12 views

A reduction of Cayley hamilton to the complex case

I found an interesting proof of the general Cayley-Hamilton theorem but i didn't quite get all of it. I'm trying to make sense of this reduction argument. I'm not that familiar with algebras so this ...
1
vote
1answer
15 views

Find a basis where vector has fixed cooridnates

What's the method for finding basis where $\beta_1=(0,2,1) \ , \ \beta_2=(1,1,2)$ and $\beta_1$ has cooridnates $1,2,-1$ and $\beta_2$ has $0,0,1$ ? we have that $(1,1,2)=(a_3,b_3,c_3)$ and ...
-2
votes
0answers
27 views

A linear system for which the solution space is spanned by the given vectors [duplicate]

Make a system with $3$ equations and $3$ unknowns of which the solution space $V$ is spanned by the column vectors: $$\begin{bmatrix} 1 \\0 \\-1\end{bmatrix},\quad \begin{bmatrix}1 \\3\\ ...
0
votes
0answers
22 views

What is recommended for studying Linear Algebra? [duplicate]

I don't know anything about linear algebra and want to start afresh but in a proper mathematical manner. What should I do and which are recommended? I know the basics of multi-variable calculus, some ...
0
votes
0answers
18 views

Find a family of linear inequalities for which the unit ball is a solution set

Find a family of linear inequalities for which the unit ball $A = \{x \in \mathbb R^n \ | \ \|x\|_2 \le 1\}$ is a solution set Would it just be $x_1^2 + ... + x_n^2 \le 1$?
2
votes
2answers
92 views

Algebraic operations with tensor products and direct sums.

When first learning about tensor products and direct sums of vector spaces, the analogy of multiplication and addition of vector spaces is sometimes used to help with the intuition. If we look at the ...
2
votes
0answers
9 views

Help with an Analysis problem involving isomorphisms and volumes.

I would like some help solving the following problem: Let $f: U \subseteq \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ be a class $C^{1}$ function. Suppose that for some $a \in U$ the derivative of $f$ ...
0
votes
1answer
24 views

prove linear independence of polynomials

Let $f_1,...,f_k$ be polynomials at field $K$ not including the zero polynomial and $\deg f_i \neq \deg f_j$ for every $i \neq j$ Show that polynomials $f_1,...,f_k $ are linear independent. I don't ...
0
votes
1answer
28 views

A Challenge on linear functional and bounding property

I took a midterm exam and after that wrote this problem down. My instructor was unable to solve it. The problem is copied here in order for anyone to help me. Suppose $f:E\to \mathbb{C}$ is a ...
1
vote
2answers
32 views

Find all 2 x 2 skew-symmetric matrices A [on hold]

Find all $2 \times 2$ skew-symmetric matrices $A$, if any, such that $A^2 + I_2 = 0$ Please help me! Thanks!!
1
vote
1answer
32 views

Give a general formula in terms of $n$ for the determinant of the following matrix.

Let $M_n$ denote the $n$ x $n$ matrix over $\mathbb{R}$ of which the entry in the $i$-th row and the $j$-th column equals $1$ if $|i-j|\leq 1$ and $0$ otherwise. For example: $M_6=$ \begin{pmatrix} ...
0
votes
0answers
10 views

Vectors transformation without using a linear regression or a neural network

Is there any way to do a regression (transforming a set of high dimension vectors ($dim=400$, $number=500$) from a state $A$ to a state $B$) without using a linear regression or a neural network. I ...
0
votes
0answers
9 views

Prove convexity of function over space of positive definite matrices

I want to show that the function $f(X) = -log \ det(X)$ is convex on the space $S$ of positive definite matrices. What I have done: It seems like this problem could be tackled by considering the ...
0
votes
0answers
14 views

Algebraic expression of a regression matrix

Let's say I'm doing a multivariate regression between a set of input $n$-dimension vectors (noted by the matrix $X=\{X_1,X_2,...X_m\}$) and a transformed version (noted by the matrix ...
0
votes
1answer
24 views

For $n>1$ $\hat x$, $p$ & $P$ [on hold]

For $n = 1$, $$\hat x = \frac{a^Tb}{a^Ta}, \quad p = a\frac{a^Tb}{a^T a}, \quad P = \frac{aa^T}{a^Ta}$$ What are these formulas when $N>1$?
4
votes
4answers
93 views

$Ax=b$ what does solving it mean?

We have been going through how to solve the system of equations known as $Ax=b$. Where $A$ is a matrix, $x$ is a vector and $b$ is a vector. I understand that if we have $A$ and $b$ we must find out ...
0
votes
2answers
29 views

be J be a matrix so: JJ^-1 = I and A a matrix so: A^t JA = J. prove that A is invertible so that AA^-1 = I

be J be a matrix so: $JJ^{-1}$ = I and A a matrix so: $A^tJA^ = J$. prove that A can invertible so that $AA^{-1} = I$ the big question here is: what are the properties of A transpose, that allows ...
1
vote
1answer
14 views

A is normal if and only if every matrix unitarily equivalent to A is normal

I need to prove that if $A$ and $B$ are unitarily equivalent, then $A$ is normal if and only if $B$ is normal. The proof is as follows: Suppose $A$ is normal and $B = U^*AU$, where $U$ is unitary. ...
1
vote
1answer
44 views

How to find the basis?

So if there is a n-dimensional space (x_1, x_2, ..., x_n) over a field, How can I find a basis which together with (c_1, c_2, ..., c_n) (where all c_i's are non-zero constants from this field) forms a ...
0
votes
1answer
29 views

tangent space of manifold and Kernel

I want to understand the proof of this Theorem : Theorem: Let $ S $ be a submanifold of $ E $ and let $ a \in S $. Assume that there exists a submersion $f : E \supset U \rightarrow F$ of class $ ...
1
vote
1answer
23 views

Proving or disproving the existence of a vector space

Suppose the vector space $V$ is spanned by $(1,0,1,0), (0,1,0,1), (1,1,0,0)$. Is it possible to find a subspace U of $\mathbb R^4$ such that $V \subsetneq U \subsetneq \mathbb R^4$? Note that ...
1
vote
1answer
26 views

Derivation of Simple Projectile Motion with Drag

Given the initial velocity $v_0$ and angle $\theta$ of a projectile on the ground, using Newton's second law and the acceleration due to gravity $\mathbf g=\left\langle0,-g\right\rangle$, I was able ...
1
vote
2answers
12 views

iid Gaussian random matrix $A\in M_n$ has full rank with probability 1?

I want to prove that: iid Gaussian random matrix $A\in M_n$(I mean whose elements are iid Gaussian) has full rank with probability 1 Below is my consideration: $$1-P(\text{full ...
0
votes
3answers
18 views

Diagonalize Matrix A

I am told to diagonalize Matrix A i solved for P and P inverse ...
0
votes
1answer
43 views

How to take the derivative of a matrix with respect to itself?

Could someone please explain how to take the derivative of matrix with respect to itself? $$\frac{\partial \textbf{X}}{\partial \textbf{X}}$$ where $\textbf{X}$ is an M x N matrix
0
votes
0answers
20 views

If the Unit Vectors are equal, Are the directions equal too?

Given that the unit vector of x = unit vector of y, can we conclude that the Direction (or Sign) of x is always equal to Direction of y?
0
votes
1answer
16 views

How do I construct a Schauder basis of $l^2$ with the set $\{v_n\}

How do I construct a Schauder basis of $l^2$ with the set $\{v_n\} \subset l^2$, where $v_n=\frac{1}{\sqrt{2}}(e_n-e_{n+1})$ if $n$ is odd and $v_n=\frac{1}{\sqrt{2}}(e_n+e_{n-1})$ if $n$ even. Note ...
2
votes
1answer
38 views

How was step 1 done in Gaussian Elimination?

Suppose I have matrix $B:= \begin{bmatrix}4 & -2 & 2\\-2 & 5 & 3\\ 2 & 3 & 7 \end{bmatrix} $ Performing Gaussian Elimination we get: EDIT corrected mistake. I mistakenly ...
2
votes
1answer
17 views

Is the function continuous that maps a vector to the coefficients of its expansion in a basis?

Let $V$ be a finite-dimensional vector space with basis $e_1, e_2, \ldots, e_n$. Consider the function $f:V\rightarrow\mathbb{R}^n$ defined such that, for each $v\in V$, $f(v)=(a_1,a_2,\ldots,a_n)$ ...
0
votes
1answer
22 views

an additive bijective map

Let H be a Hilbert space and $\Phi:B(H)\longrightarrow B(H)$ is an additive bijective map. If $\mathbb{R}I⊆\Phi(\mathbb{R}I)$, can we conclude by the bijectivity of $\Phi$ that ? ...
0
votes
0answers
18 views

Vector Algebra Coordinate Transformation

Let us look at two coordinate systems $K$ and $K'$ with axes, respectively, $(x_1,x_2,x_3)$ and $(x_1',x_2',x_3')$ and unit vectors ($\vec{e_1},\vec{e_2},\vec{e_3}$) and ...
0
votes
0answers
19 views

If $\vec{v}$ is any non-zero vector perpendicular to $\vec{u}$, show that $\vec{v}$ is an eigenvector of $S$ [on hold]

I have the following problem.. I solved the first one however i can't find out how to solve the second (b). Suppose $\vec{u}$ is a unit row-vector in $\mathbb R^n$ , and $A=uu'$ matrix. (a) ...
1
vote
3answers
59 views

$x-y-2z=0$ find a perpendicular vector

Why is the vector $e=(1,-1,-2)$ ?
2
votes
1answer
27 views

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$ Having no success with this question, I turn for your help =] I ...
1
vote
1answer
33 views

Bound spectral radius of a certain matrix.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi = \text{diag}(\xi)$ to be a diagonal matrix with a principal left eigenvector $\xi$ of $P$ (i.e. ...
-2
votes
1answer
31 views

The dimension and basis of the set $F = \{(a+3b,a-b,2a-b,4b)| a,b \in \mathbb{R}\}$

Let $F = \{(a+3b,a-b,2a-b,4b)| a,b \in \mathbb{R}\}$ Show that F is a subspace of $\mathbb{R}^4$; Find a basis for F; Find the dimension of F. I have part A completed and showed ...
2
votes
0answers
35 views

showing something is an inner product

I'm trying to do the question above. I have found $f(v,w) = v^TAw$ where $A = \begin{pmatrix} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 ...
0
votes
0answers
16 views

Find sum of vector formula (?)

Find the sum of $$x(y > 10) + z(y < 30),$$ where $x$, $y$ and $z$ are vectors and $$x(y>10) = \{x_i: y_i > 10\}.$$ I have no idea where to even get started on this problem. ...
1
vote
2answers
28 views

What is the meaning of these summations?

Am I meant to add the two summations together, or multiply them? If the latter, what makes it any different from an outer product?
0
votes
2answers
53 views

Is the zero matrix diagonalizable?

Then for any invertible matrix $P$, we can say $P^{-1}\cdot 0 \cdot P=0$ ?
0
votes
3answers
27 views

Finding subspace's base

Let W be a subspace of $\mathbb{R}^4$: $ \begin{cases} x_1+2x_2+3x_3+4x_4=0 \\ 2x_1+2x_2+x_3+3x_4=0 \end{cases}$ Find base of W and extend it to the base of $\mathbb{R}^4$ How to approach this ...
1
vote
1answer
38 views

Misshap/typo in question/answer?

I am trying to understand question 3. The first image below is some information about how to solve it. Take notice of the of marked yellow area, espicially about how it becomes $1/9$. Here on ...
0
votes
1answer
25 views

Generalized Eigenvector for 4x4 matrix

I'm working on Systems of Differential Equations and I'm looking to find the Generalized eigenvector for the following matrix: $\left[\begin{array}{rrrr} 3 &-4 &1 &0 \\ 4& 3 &0 ...
0
votes
1answer
34 views

How to rewrite a derivative w.r.t. tensor as w.r.t. vector

I'm stuck on a (probably very simple) problem I've come across. Take a function $f(A)$ where $A$ is a 2-tensor. Now suppose $A=vv^T$ for an $\mathbb{R}^n$ vector, $v$. I want to rewrite the object ...