1
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1answer
47 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
38 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
86 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
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1answer
32 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
35 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
29 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
20 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
35 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
191 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 $m=0$, ...
0
votes
1answer
136 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
69 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
63 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
29 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
293 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
33 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
35 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
32 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
29 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
21 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
52 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
48 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
66 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
59 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
39 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
144 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
20 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
47 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
84 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
68 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
86 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 ...
4
votes
2answers
80 views

Linear Algebra: Direct Sum

Prove that if $W_1$ is any subspace of a finite-dimensional vector space $V$, then there exists a subspace $W_2$ of $V$ such that $V = W_1 \oplus W_2$ What I have done so far is to note that ...
2
votes
4answers
149 views

Prove that the vectors $v_1,v_2,\ldots,v_k \operatorname{span}R^n$ if and only if $[v_1]_B,[v_2]_B,\ldots,[v_k]_B \operatorname{span}R^n$.

From section on Change of Basis $\longrightarrow$ Assume the vectors $v_1,v_2,\ldots,v_k\operatorname{span}R^n$, we must show that $[v_1]_B,[v_2]_B,\ldots,[v_k]_B\operatorname{span}R^n$. We can ...
0
votes
1answer
27 views

Show that the entries of a matrix are:

For a regression model $y=\beta x$ (note there is no intercept term), show that entries of the matrix $\bf{H} = \bf{X}[\bf{X'}\bf{X}]^{-1}\bf{X'}$ are $h_{ij} = ...
0
votes
2answers
48 views

Sum of the eigenvalues

if $V$ is a finite-dimensional vector space and $t \in \mathcal L (V,V) $is such that $t^2 = id_V$ prove that the sum of eigenvalues of t is an integer. I started the prove as such: Let $\lambda_1 ...
2
votes
2answers
63 views

Could this linear algebra proof be done without computation?

From page 95 of Hoffman & Kunze's Linear algebra: Let $T$ be the linear operator on $\mathbb{R}^2$ defined by $T(x_1,x_2)=(-x_2,x_1)$ Prove that if $B$ is any ordered basis ...
0
votes
1answer
70 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by finding a linear function T

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
3
votes
0answers
69 views

Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
0
votes
2answers
39 views

Algebraic proof with matrices

I need to proof the following: Given $A$ is a $n\times n$ matrix so that $A^2 - 3A + I = 0$ Prove that $A^{-1} = 3I - A$ So I laid out a matrix: $$ A =\begin{pmatrix} a & b \\ c & d ...
1
vote
1answer
99 views

Prove: If A is invertible, then adj(A) is invertible and $[adj(A)]^{-1}=\frac{1}{det(A)}A=adj(A^{-1})$

I can show the left side: $$A^{-1}=\frac{1}{det(A)}adj(A)$$ $$AA^{-1}=\frac{1}{det(A)}A*adj(A)\longrightarrow I=\frac{1}{det(A)}A*adj(A)$$ and, $$A^{-1}A=adj(A)\frac{1}{det(A)}A \longrightarrow ...
1
vote
1answer
35 views

Direct sum of $3$ subspaces

$V_1$,$V_2$,$V_3$ are subspaces of vector space $V$. How to prove that if $V_1 \cap \left(V_2+V_3\right) = V_2 \cap \left(V_1+V_3\right) = V_3 \cap \left(V_2+V_3\right)=\{0\}$ so $V_1\oplus V_2 ...
2
votes
3answers
81 views

Proving that $\lambda$ being an eigenvalue for $A$ implies $\lambda^{-1}$ is an eigenvalue for $A^{-1}$

Let $A$ be an invertible matrix, and let $\lambda$ be an eigenvalue for $A$. We have that $Ax = \lambda x$ for some eigenvector $x$. Note that $A^{-1}Ax = A^{-1}\lambda x$, which gives $x = ...
0
votes
1answer
35 views

Vector spaces: Isomorphism and non-singular matrices

The following true/false question was posed: An isomorphism between to vector spaces can always be represented by a square singular matrix. This is not true. I know that in the case of finite ...
3
votes
1answer
68 views

Linear Algebra: Identity map

I was asked to prove that the identity map $id : \Bbb R^n \to \Bbb R^n $ can be represented by the the identity matrix regardless of the basis My Attempt: Let $\mathcal B = \lbrace v_1 , ...,v_n ...
2
votes
0answers
58 views

Proof of two properties of a simple math function

I would like to define a function to evaluate the value for some entities which receive a number of "up"s ($\mathcal{u}$) and "down"s ($\mathcal{d}$). I devised the following function: ...
0
votes
1answer
78 views

The annihilator of an intersection

I know this question has been arlready asked, but as my reputation is too low I'm not allowed to post a comment, sorry for this second post. I'm asked to prove : $(W_1+W_2)^0=W_1^0\cap W_2^0$. ...
6
votes
5answers
348 views

Proof Strategy - Prove that each eigenvalue of $A^{2}$ is real and is less than or equal to zero - 2011 8C

Remember that we've already proven the following, for any real symmetric $n\times n$ matrix $M$: (i) Each eigenvalue of $M$ is real. (ii) Each eigenvector can be chosen to be real. (iii) Eigenvectors ...
2
votes
3answers
108 views

Prove $\det(A)=\det(A^T)$ detail

I want to prove that $$\det(A)=\det(A^T)$$ and the one step I don't understand (the problem is guiding you thought it is to prove $$P^T_{\sigma} = P_{\sigma^{-1}}$$ where P is a permutation matrix. ...
0
votes
1answer
67 views

Differential map on the vector space of polynomials: Kernel and Image

Given the $V_n$ is the vector space of polynomials of degree $\leq$ n over $\Bbb R$ So $M_D = \begin{bmatrix} 0 & 1 & ... & 0\\ 0 & 0 & 2 &... 0 \\ 0 & 0 &0 &.\\ . ...