0
votes
1answer
28 views

Linear Transformation from V to W (bijective) Show that T(v) is a basis of W if B is a basis of V.

$V, W$ two vector spaces and $T: V \to W$ is a bijective linear transformation. $B$ is a basis of $V$. Prove that $\{T(\mathbf{v}) | \mathbf{v} \in B\}$ is a basis of $W$. I started by doing ...
1
vote
3answers
99 views

Strange proof of Schwarz Inequality with Pythagorean Theorem

Does anyone know what is going on in this proof of the Schwarz inequality? Most importantly: how can one assume that $c^2\leqq \|A\|^2$, or later on, that $c^2\|B\| \leqq \|A\|^2$? This would imply ...
3
votes
2answers
39 views

Proving $v$, $T(v)$, $T^2(v)$ is a basis

I'm trying to prove the following: Given that $V$ is a vector space, with $dim V = 3$, and $T: V \to V$ is a linear map with the properties $T^2(v) \neq 0$ and $T^3(v) = 0$, with $v \in V$, show that ...
1
vote
0answers
34 views

Can you check my proof concerning an invariant subspace under a diagonilizable linear operator and its complementary invariant subspace?

This was an exercise problem from H&K Linear Algebra(sec 7.2, exercise 18). Could you check my proof? The theorem is as follows: If $V$ is a finite-dimensional vector space and $W$ is an ...
3
votes
1answer
43 views

Commuting quaternions

I tried to solve the following exercise, please could somebody tell me if I did it right?: Prove that non-real elements $x,y \in \mathbb H$ commute if and only if their imaginary parts are ...
1
vote
1answer
112 views

Proof Verification, Uniqueness of vector $v$ satisfying $\varphi(u) = \langle u,v\rangle $ for a linear functional $\varphi$.

I want to prove the uniqueness of the following vector $v$. The existence of the vector is guaranteed. We know that there exists at least one vector $v$ for every $u$ such that for a linear ...
0
votes
1answer
29 views

Proof about linear systems of equations

If $X_1$,$X_2$ are solutions of $AX=B \neq 0 $ then $aX_1 + bX_2$ is never a solution. I tryed this way: From the hypotesis we have $AX_1=B$ and $AX_2=B$ with $B \neq 0$. Then: $A(aX_1 + ...
0
votes
1answer
25 views

Proof Writing Help: $P_UT=TP_U \Leftrightarrow U$ and $U^{\perp}$ are $T$-Invariant

I'm studying linear algebra using Axler's book on my own and this is also my first rigorous encounter with proofs would greatly appreciate suggestions to improve the writing of the first part of my ...
2
votes
3answers
90 views

Center of $GL_n(\mathbb R)$ is the set of matrices $\lambda I$

I determined the set of all matrices $A$ such that $AB = BA$ for all $B$ in $GL_n(\mathbb R)$ to be the set of $\lambda I$. Now I'm not sure this is true. But quite sure. So I tried to prove it and it ...
2
votes
2answers
60 views

Prove $\dim W \ge 2$

Let $U_1, U_2, W$ subspaces of a finite dimensional vector space, such that: $U_1 \cap U_2 = \{0\}$ $U_1 \cap W \ne \{0\}$ $U_2 \cap W \ne \{0\}$ Show that $\dim W \ge 2$. ...
0
votes
0answers
36 views

Matrix of a linear application

Given the vectors $ v_1=(-1,2,-3)$,$v_2=(0,1,1)$,$ v_3 = (0,1,-1)$,$v_4=(1,1,4)$ and $ w_2 = (3,-1,2)$,$w_3= (1,-1,0)$ , $ w_4 = (t,-3,4)$ a) Find for which value of $t$ there is a linear application ...
3
votes
2answers
37 views

Show $\cos\theta=\frac12(\text{tr}(g)-1)$ with $g\in\text{SO}(3)$

How can I show that for $g\in\text{SO}(3)$ given by $\begin{pmatrix}1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta\end{pmatrix}$ the equality ...
0
votes
3answers
47 views

Proving commutativity of addition for vector spaces

I'm trying to prove commutativity of addition for vector spaces, using the axioms for vector spaces. Apparently commutativity can be proven! Im having trouble getting a good feel for what is allowed ...
2
votes
1answer
39 views

$V$ is a linear space. Need to compute $T^n$

Given: $V$ is a linear space $(\dim V = n)$ and there is a linear transformation $T: V \rightarrow V$ that $T^n = 0$ and $T^{n-1} \ne 0$ , also there's $u \in V$ that $T^{n-1}(u) \ne 0$ Prove ...
1
vote
0answers
32 views

Finding the $\text {Im } (f^2)$

Let $f: \mathbb{R}^3 \to \mathbb{R}^3$ defined by $f(a,b,c)=(c-b,a-c,b-a)$ be a linear application. The matrix of $f$ is $A=\begin{bmatrix} 0 & -1 &1 \\ 1&0&-1\\-1&1&0 ...
0
votes
2answers
41 views

VERIFYING: Proving an $n \times n$ Matrix Vector Space

Let $V$ be the vector space made by the $n \times n$ square matrices. a) Prove that $S=\{A\in V|A^t=A\}$ is a subspace of $V$ b) Prove that $T=\{A\in V|A^t=-A\}$ is a subspace of $V$ c) Prove that ...
1
vote
1answer
16 views

Prove $\{(x_1 ,x_2, 0) : x_1, x_2 ∈ F\}$ is a subspace of $F^3$.

$(x_1 ,x_2, 0) + (y_1, y_2, 0) =((x_1 + y_1), (x_2 + y_2), 0)).$ So, it's closed under +. $a(x_1 ,x_2, 0) = ax_1, ax_2, 0$. So, it's closed under *. Vector $(0, 0, 0) \in \mathbb F^3$ and its ...
2
votes
1answer
81 views

Proof that a is an eigen value of p(T) if and only if a=p(lambda) for some eigenvalue lambda of T

$\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\C}{\mathbb{C}} \newcommand{\LM}{\mathcal{L}}$ Question: Suppose $\F = \C, T \in \LM(V), p \in ...
0
votes
0answers
22 views

is this a valid counter-example - function is not locally invertible

Let $S_n$ be the set of all symmetric matrices with real entries of size $n$x$n$. We are asked if the function $f:S_n \to S_n$, $f(A)=A^2$ is locally invertible for every $A$ (Using the Inverse ...
4
votes
1answer
31 views

Using inner product property to determine if operator is an isomorphism.

Let $\varphi$ be an operator on a $k$-vector space $V$ with an inner product $\langle\cdot,\cdot\rangle$. Suppose that $\langle v,\varphi v\rangle = 0$ for every $v\in V$. If we take $k=\mathbb R$, is ...
4
votes
6answers
109 views

Find maximal possible determinant value given constraint

Task is to find maximal possible determinant value for 2x2 and 3x3 matrices given following constraint: $$\sum_{i,j=1}^na_{ij}^2 \le 1$$ I was able to come up with solution, but I received the test ...
1
vote
1answer
40 views

Confused about a particular example of rational canonical form… please help me find my error.

The minimal polynomial of the matrix $A = (\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix})$ is $x^2 + 1$. (At least, I think so - how can one be sure about this?) If we think of this ...
0
votes
0answers
11 views

Verification of step in a proof of the decomposition of primary f.g torsion modules over PIDs?

I was reading about the decomposition of finitely generated primary torsion modules over PIDs, and though of an alternative way to do the "inductive" step. Since it is substantially simpler than the ...
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) : ...
0
votes
0answers
21 views

Matrix rank and linear independence

$\mathbf{a}_i,\mathbf{b}_j$ are $n$ dimensional vectors. Consider the matrix $\mathbf{M}$ defined by: $$\mathbf{M}_{ij}=\mathbf{a}_i\cdot\mathbf{b}_j$$ Prove/disprove that ...
2
votes
1answer
44 views

A question about Matrices and Linear Transfromations

Let $v_1,...,v_n$ be a basis of a vector space $V$ over a field $K$. Let $M(T)$ denote the matrix of a linear map $T:V \rightarrow V$ with respect to our basis. Prove $$M(ST)=M(S)M(T)$$ for all ...
0
votes
1answer
41 views

If $g:V \rightarrow V$ is an injective linear transformation. Prove if $V$ is finite dimensional then $g$ is surjective.

I am asked to prove this without the rank nullity theorem My Attempt at a Proof For the $\implies$; If $g:V \rightarrow V$ is injective then the dimension of the kernel is 0, and so as ($im$) ...
2
votes
1answer
116 views

Check if the following gradient is correct

This question regards the verification of the gradient of a given function. Notation. Let $N, K \in \mathbb{N}_0$ be given (nonzero) integers, with $K > N$. Let $\mathbf{x} = [x_b \ y_b \ z_b]^T ...
1
vote
1answer
53 views

Find matrices complying to given constraints

We are given linear mapping of $n$-dimensional vector space, such as: It has $n+1$ eigenvectors Any $n$ of them are linearly independent Find all matrices which could define such a linear ...
1
vote
2answers
170 views

Proof of Cayley-Hamilton Theorem for Diagonalisable Matrices [Lay P326 Ch 5 Sup Q7]

Proof for Diagonal Matrices from Page 2 of 7: Let $A \in M_{n}(C)$ be diagonal, to wit, $A _{ii}=\lambda_{i}$. Then $ p_{A}(t) = \det(tI-A)= \det \begin{bmatrix} t - \lambda_1 & ~ & ~ \\ ...
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
0answers
16 views

Which is the starting and ending basis? - Matrix of a linear transformation [Lay P294 Q 5.4.28]

Denote some arbitrary linear transformation as $L.$ When a question asks "to find a matrix of $L$ with respect to S and T", does this denote $[L]_{T \leftarrow S}$ or $[L]_{S \leftarrow T}$ ? How can ...
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 ...
1
vote
4answers
56 views

I need help with a proof showing $\|u\|^2 = \|\operatorname{proj}_v u\|^2 + \|u - \operatorname{proj}_v u\|^2 $

So, I am dealing with the 2-norm and the projection is defined as the standard orthogonal projection, so far I have $$\|u\|^2 = \|\operatorname{proj}_v u\|^2 + \|u - \operatorname{proj}_v u\|^2 ...
3
votes
2answers
24 views

Rank-one perturbation proof

I wrote a proof for a problem in my textbook. Can someone please verify it or offer suggestions for improvement? $\textbf{Problem:} $If $u$ and $v$ are $m$-vectors, the matrix $A = I+uv^*$ is known ...
2
votes
1answer
30 views

Question about the operator norm on $\mathbb R^2$

So here is my question, I have to decide whether the following statement is true Let $T$ be an isomorphism on $\mathbb R^2$. Then $$\|T\|=\frac{1}{\|T^{-1}\|}$$ I am pretty sure that the statement ...
6
votes
0answers
73 views

A power of the characteristic polynomial

Let $A$ be a square matrix with real or complex coefficients of size $n$. Define its characteristic polynomial by $\chi_A(X) = \det(A-XI_n)$ (or $\det(XI_n-A)$ if you prefer). The question is : Prove ...
4
votes
3answers
82 views

Verifying that the determinant is equal to $1!2!3!…(n-1)!$

Verifying that the determinant is equal to $1!2!3!...(n-1)!$ $$|A|= \begin{vmatrix} 1 &1 & \dots &1\\ 1 &2 & \dots &2^{n-1}\\ 1 &3 & \dots &3^{n-1}\\ & & ...
4
votes
4answers
711 views

Showing AB=0 does not imply either A,B=0, but that singular

Ex. 8.5 - Mathematical Methods for Physics and Engineering (Riley) By considering the matrices $$ A = \left( \begin{matrix} 1 & 0 \\ 0 & 0 \\ \end{matrix} \right) ...
1
vote
2answers
43 views

Why is this proof complete if only one condition is satisfied?

In the 10th ed. of Elementary Linear Algebra (Anton), the following statement exists: If Ax = b is a system of linear equations, exactly one of the following is true: (a) the system has no ...
1
vote
1answer
67 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 ...
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 $p(\lambda)=p_{0}+p_{1}\lambda+\ldots+p_{n-1}\lambda^{n-1}+p_{n}\lambda^{n}$. ...
3
votes
1answer
67 views

What's wrong with $\det(P) = -1$ : Change of variable for Quadric Forms ? [Kolman P552 8.7.25]

Would someone please explain "why $\det(P) = 1$ is required" and the general procedure of effecting this? Lay S7.2 didn't expound on this and neither does Kolman in S8.6-8.8. Identify the graph ...
5
votes
1answer
48 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
43 views

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W This is something from a practice sheet I got. I'm studying for a linear algebra final. I am unsure if we have ...
3
votes
1answer
50 views

Show: $\varphi\colon\mathbb{Z}_{mn}\to\mathbb{Z}_m\times\mathbb{Z}_n, k\mapsto (k\% m,k\% n)$ is a ring isomorphism for $m$ and $n$ relatively prim

Let $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Let $m,n\in\mathbb{N}$ be relatively ...
0
votes
2answers
44 views

If T(S) is linearly independent, show S is linearly independent

Let $T: V \to W$ be a linear transformation. Let $S = \{v_1,...,v_k\}$ and assume $T(S)$ is linearly independent. Show S is also linearly independent. I think I just have to prove that if $a_1 v_1 + ...
16
votes
10answers
1k views

Assume that the square matrix A has an eigenvalue of 0. Is A invertible? Why or why not?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof:Suppose $A$ is square and invertible and for the sake of contradiction let $0$ ...
2
votes
1answer
108 views

Valid Proof for Cayley Hamilton Theorem? (Not the usual incorrect one)

By induction; case n=1 is true. $A$ admits an eigenvalue $\lambda$ with eigenvector $v$ over $\mathbb{C}$. Change $A$ into a basis $e_1=v,...,e_n$. Then $\exists X$ such that $XAX^{-1}=\left( ...