0
votes
1answer
16 views

Prove If a set contains more vectors than there are entries in each vector, then the set is linearly dependent

I want to prove this theorem: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set $\{ v_1,v_2,...,v_p \}$ in $\mathbb{R}^n$ is ...
1
vote
0answers
21 views

Check my proof - Linear Algebra

Still not completely confident with my capabilities in writing formal proofs so I thought I would ask for a check of this proof. Theorem Let $V$ and $W$ be vector spaces, and let $T$ and $U$ be ...
0
votes
1answer
50 views

“The Conjugate of a matrix”

I am having some trouble understanding a definition/question in my linear algebra text book. The question states " If $A$ is a square matrix, a matrix of the form $P^{-1}AP$ where $P$ is invertible ...
1
vote
2answers
23 views

induction for idempotent matrix : $P^n = P$

Given that $P^2 = P$ how do i prove by induction that $P^n = P$? I have tried the following: we know that $P^k = P$ holds for $k = \{1,2\}$. If we now take $k=3$: $$ \begin{align} P^3 &= ...
0
votes
0answers
10 views

show that for the system to be consistent we must have b2 = b3 - 2b1 [closed]

The first problem, #4. I dont know how to approach this problem, I can see how the statement b2 = b3 - 2b1 is true but how do I prove it must be true?
1
vote
1answer
25 views

Verify that $(I−XY)^{(-1)}*X=X*(I−YX)^{(-1)}$ [duplicate]

Verify that $(I_n−XY)^{-1}\cdot X=X\cdot (I_m−YX)^{-1}$ The first $I$ is of order $n$ and the second is of order $m$. $X$ is $n\times m$ $Y$ is $m\times n$
1
vote
1answer
153 views

Linear algebra proof regarding matrices

I'd like a hint rather than a full solution. The problem I am considering is the following: $X$ is an $n\times m$ matrix $Y$ is $m\times n$ Show that $(I - XY)^{-1}\cdot X = X\cdot(I - ...
0
votes
3answers
49 views

Proofs about Matrix Rank

I'm trying to prove the following two statements. I can prove them easily by considering the matrix as a representation of a linear map with a given basis, but I don't know a proof which uses just the ...
0
votes
2answers
58 views

General vector space theory developed without matrix-theory.

Since vector spaces can exist regardless of a matrix I wanted to see if we could do all the proofs for the general vector-spaces without using theory for matrices. Then it was only two proofs of the ...
0
votes
1answer
28 views

Prove that $\dim(U_{\perp}) = \dim(V ) − \dim(U)$.

Let $V$ be a finite-dimensional inner product space over field $F$, and let U be a subspace of $V$ . Prove that the orthogonal complement $U_{\perp}$ of $U$ with respect to the inner product $\langle ...
1
vote
0answers
46 views

How to find intelligently counterexamples for (dis)proofs about matrices?

Let's say I'm asked to give a counterexample for a claim about matrices, such as The elementwise product of two positive semi-definite matrices is positive semi-definite. It's easy enough to do ...
0
votes
1answer
28 views

Proving $\mathrm{Hom}(V \rightarrow W)$ is a vector space

It can easily be proven that $\newcommand{\Hom}{\mathrm{Hom}}\Hom(V \rightarrow W)$ is a sub-space. 1. we know that for any $T:V\rightarrow W$, T(0)=0, therefore $0\in \Hom(V \rightarrow W)$ 2. ...
0
votes
0answers
19 views

Every polyhedron $P \ne \mathbb{K}^n$ equals an intersection of finitely many half spaces.

Currently, I am reading some lecture notes on linear optimisation. I cannot see why the following (seemingly trivial) proposition holds. (How could I understand/proove it?) Every polyhedron $P \ne ...
0
votes
1answer
27 views

Prove: If this system is solvable, then this dual system is not.

I'm trying to get a handle on algebraic dual spaces, and it's hurting my head. To be proven: Let $A$ be a $m \times n$-matrix and $b$ be a $1 \times n$-matrix. Show that the system $$\begin{cases} ...
0
votes
1answer
39 views

How to prove the uniqueness of linear functional

$\textbf{Theorem}$ If $V$ is a $n$-dimensional vector space, if $\{x_1,.,.,., x_n\}$ is a basis in $V$ and if $\{\alpha_1,\cdots \alpha_n\}$ is any set of $n$ scalars, then there is one and only one ...
2
votes
0answers
35 views

Proof of Gersgorin Discs Theorem

I have a question about the proof of Gersgorin Theorem from the book Matrix Analysis by Horn & Johnson. The Theorem states that for any $A\in \mathbb{C}^{n \times n}$ 1) all eigenvalues are ...
1
vote
2answers
43 views

Doubt in proof of Dual of the direct sum

If $M$ and $N$ are subspaces of $V$, and if $V = M \oplus N$, then $$V' = M^\perp \oplus N^\perp$$ where $W^\perp$ is the annihilator of $W$. I didn't understand how to prove both of the ...
3
votes
3answers
50 views

Show surjectivity of a linear map

It pains me to say that this bewilders me, but here's the problem. All I want to do is show that: Given $T$ a linear operator on some finite-dimensional space $V$, with the property that $Im(T) = ...
1
vote
1answer
48 views

Is this sufficient for linear independence proofs??

I've been doing all of these proofs the same basically, I just want to make sure I'm doing them right, I didn't include all the details but I have the outlines of my proofs here. 1) U and W are ...
1
vote
2answers
42 views

Invertible Proof with transposed matrices

Let A, B, C be square matrices that are invertible. Say I want to express X with no inverses Say $$ (A^{T}A)^{-1}(X +B^ {T})(C^{-1}B^{-1})^{T} = I. $$ I know that $A^{T}A$ = $I$, but where can I go ...
0
votes
3answers
98 views

Sum of invertible matrices proof

If we have two square matrices, $A$ and $B$. Assume that $A + B$ is invertible. Would that mean that $A^{-1} + B^{-1}$ is invertible too?
1
vote
1answer
38 views

Proof that P is an Orthogonal Projection

I'm studying linear algebra using Axler's text and am stuck on 6.17. The problem is: Prove that if $P \in \mathcal{L}(V)$ is such that $P^2 = P$ and every vector in $\operatorname{null} P$ is ...
3
votes
2answers
40 views

Basis and dim of the set of all $n\times n$ symmetric matrices.

An $n \times n$ square matrix $A$ is called symmetric if $A^T = A$ Show that the set of all $n \times n$ symmetric matrices, denoted $S$, is a subspace of $M_n(\mathbb{R})$. Give a basis for $S$ ...
0
votes
0answers
31 views

Matrix Saddle Points and Dominance

I was teaching myself about dominance relations and saddle points after a friend of mine started discussing it with me and how it can be used in games. I wanted to know how to prove a problem like ...
0
votes
1answer
56 views

Geometric proof of dot product distributive property

I'm working my way through a text book for fun in order to keep my math brain fresh and came across this simple yet perplexing problem. "Demonstrate geometrically that the dot product is ...
1
vote
1answer
37 views

Endomorphism with rank $r$ annihilates degree $r+1$ polynomial

Let $f$ be a linear transformation of $\mathbb R^n\to \mathbb R^n$ that has rank $r$. Prove the existence of a degree $r+1$ polynomial that annihilates $f$ I have a proof : consider $g$ ...
4
votes
4answers
223 views

Prove that p has m distinct roots if and only if p and p' have no roots in common

Problem: Suppose $p \in \mathcal{P}(\mathbf{C})$ has degree $m$. Prove that $p$ has $m$ distinct roots if and only if $p$ and its derivative $p'$ have no roots in common. My proof so far: If $m=0$, ...
1
vote
1answer
149 views

Find bases given that P is the change of coordinates matrix from this to this [Lay P244 Q4.7.19]

Lay P289: Let $V$ be an $n$-dimensional vector space, let $W$ be an $m$-dimensional vector space, and let $T$ be any linear transformation from $V$ to $W$. To associate a matrix with $T$, choose ...
0
votes
1answer
72 views

Any $2\times 2$ complex matrix A is similar to one of these three: (See first line of the question)

(i) : $\left(\begin{array}{ll} \lambda_{1} & 0\\ 0 & \lambda_{2} \end{array}\right)$, (ii) : $\left(\begin{array}{ll} \lambda & 0\\ 0 & \lambda \end{array}\right)$, (iii) : ...
1
vote
2answers
67 views

What's the fastest way to determine Eigenvalues & Eigenvectors of any 2 by 2 Matrix?

My instructor claims that it's inefficient and superfluous to compute eigenvectors de novo for each $2$ by $2$ matrix. He suggested a trick instead which resembles the eigenvectors and cases here. ...
0
votes
1answer
32 views

Rank Nullity and Dimension relation

How would one prove the relations: $rank S◦T = rankT-dim(kerS ∩ ImT)$ and $nullity S◦T = nullityT+dim(kerS ∩ ImT)$ I understand that the use of rank nullity theorem is required but am confused by ...
2
votes
2answers
297 views

When Dim eigenspace = 1, any $2\times 2$ complex matrix A is similar to $\left(\begin{array}{ll} \lambda & 1\\ 0 & \lambda \end{array}\right)$.

$\bbox[5px,border:2px solid gray]{ \text{ Case 3 } }$ If $\dim E_{\lambda}=1$, take a nonzero $v\in E_{\lambda}$, then $\{v\}$ is a basis for $E_{\lambda}$. Extend this to a basis $\mathfrak{B}=\{v,\ ...
1
vote
2answers
37 views

Easier Proof - Union of finite lin-indep subsets of the eigenspaces = a lin-indep subset. [Lay P285 Thm 5.3.7c]

P267 Lemma. Let $T$ be a linear opera $tor$, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. For eacb $i=1,2,\ \ldots,\ k$, let $v_{i}\in E_{\lambda;}$, the ...
1
vote
1answer
37 views

Intuition and Motivation - Linear Operator $T - \lambda_k I$ ? [Lay P270 Thm 5.1.2]

Let $T$ be a linear operator on a vector space V, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. If $v_{1},\ v_{2},\ \ldots,\ v_{k}$ are eigenvectors of $T$ ...
0
votes
1answer
38 views

Adaptation of this proof of spectral theorem to the complex case

My question is quite simple, I would like to know why we can't use this proof to the complex case, i.e., the operator $T$ is self adjoint on a complex n-dimensional inner product space $V$. Can we ...
1
vote
1answer
30 views

Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. [Lay P160 Ch 2 Sup Q4]

Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To verify this, compute $ (I \color{orangered}{-A} ...
2
votes
2answers
23 views

Mustn't both left and right inverses be checked? [Lay P160 Ch 2 Sup Q4]

Question: Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To ...
0
votes
0answers
13 views

Proof of update rule for perceptrons

$$w = \sum_n \tau^n\phi^nt^n$$ $$t^n = -1\text{ or }+1.$$ $\phi^n$ is a vector. $\tau^n$ is the number of times each vector $\phi^n$ has been presented and misclassified. It can be shown that ...
1
vote
1answer
32 views

Proof: If $F^3 = F$ then F is diagonalisable

let $V$ be a $\mathbb{R}$-vectorspace with $dim V < \infty$ and $F$ an endomorphism of V with $F^3 = F$. Show: F is diagonalisable. $F^3 = F$ is equivalent to $F^3 - F = 0$. Now I know that ...
1
vote
2answers
56 views

Proof: $\exists$ subspace $U$ of $ker(f)$ with $U \bigoplus T_1 = T_2 $

I need help with this proof: Let $V, W$ be K-vectorspaces. Let $T_1, T_2$ be subspaces of V with $T_1 \subseteq T_2$. Let $f \in hom_K(V,W)$. Show the following: If $ f(T_1) = f(T_2)$ then exists a ...
1
vote
1answer
50 views

For $W \leqslant V$, prove $\dim W + \dim W^\perp = \dim V$

I want to prove that if $W$ is a subspace of $V$, then $\dim W + \dim W^\perp = \dim V$. I have defined $x^\perp = \{ y : x \cdot y = 0\}$, where $\cdot$ denotes the dot product. It is a pretty ...
1
vote
1answer
91 views

Find an ordered basis of $V$ such that $[T]_\beta$ is a diagonal matrix.

The entire problem statement is: Let $V$ be a finite dimensional vector space and $T:V\to V$ be the projection of $W$ along $W'$, where $W$ and $W'$ are subspaces of $V$. Find an ordered basis ...
2
votes
4answers
65 views

Show if $A$ has a zero row, then $AB$ has a zero row.

Let $A$ and $B$ be $n \times n$ matrices. Show that if the $i$th row of $A$ has all zero entries, then the $i$th row of $AB$ will have all zero entries. Also give and example using $2 \times 2$ ...
3
votes
1answer
44 views

A Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues - Proof Strategy [Lay P397 Thm 7.1.3c]

Would someone please explain the proof strategy at Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues? I brook the algebra so I'm not asking about ...
7
votes
1answer
152 views

Strategy of a purely algebraic proof of Cayley-Hamilton Theorem

Let $p(\lambda)=det(A-\lambda l)$ be the characteristic polynomial of a $n \times n$ matrix $A$. Then $p(A)=O.$ Let $p(\lambda)=p_{0}+p_{1}\lambda+\ldots+p_{n-1}\lambda^{n-1}+p_{n}\lambda^{n}$. ...
0
votes
1answer
21 views

Linear Algebra Question: Prove that no proper subset spans

I have to prove that "S is a basis for linear space L if and only if it is a minimal spanning set for L. In other words S is a basis for L if and only if S spans L and no proper subset of S spans ...
5
votes
1answer
49 views

Why must P be orthonormal, and not just orthogonal, for change of variable in Quadratic Form? [Kolman P560 8.8.24]

Lay P402 : A change of variable is an equation of the form $x=Py$, where $P$ is an invertible matrix and $y$ is the (neW) coordinate vector of $x$ relative to the basis of $\mathbb{R}^{n}$ determined ...
1
vote
1answer
94 views

Cauchy-Schwarz Inequality - Proof using Quadratic Polynomial [Lay P379 Thm 6.7.16]

I don't perceive https://www.dpmms.cam.ac.uk/~wtg10/csineq.html, about why it " is an obvious thing to write down" "a quadratic form, use the fact that it is non-negative everywhere, and ...
1
vote
1answer
110 views

Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues - Proof Strategy [Lay P397 Thm 3]

Herein, I denote the Hermitian conjugate by * (ie: $A* = \bar{A}^T) $. Let $v_i$ and $v_j$ be two eigenvectors of an Hermitian matrix H. First of all suppose that their respective eigenvalues i and j ...
2
votes
1answer
107 views

Cauchy-Schwarz Inequality - Proof using Projections [Lay P379 Thm 6.7.16]

t If $u=0$, then the inequality becomes $ 0 \le 0 $, which is true. See Practice Problem 6.7.1 on P382. If $u\neq 0$, let $W$ be the subspace spanned by $u$. $1.$ How would one determine to ...