3
votes
1answer
37 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
41 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 + ...
15
votes
9answers
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
79 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( ...
1
vote
0answers
56 views

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

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 ...
0
votes
1answer
29 views

Question about eigenvalues: eigenvalue $f^2 + f = -1 \rightarrow$ eigenvalue $f^3 = 1$

I have to proof: Let $f \in End(V)$. Show that if $f^2+f$ has eigenvalue $-1$ then $f^3$ has eigenvalue $1$. My idea: If $-1$ is the eigenvalue of $f^2+f$ then there exists (per definition) a $v ...
0
votes
1answer
28 views

Projection operator image and kernell

Proof that: Linear operator ($P:X\to X$) is projective ($P^2=P$) IFF $\exists$ direct sum decomposition of X ($X=V\oplus U$), such that $\forall u\in U:Pu=u$ and $\forall v\in V:Pv=0$. My proof: ...
2
votes
3answers
53 views

Determine if the set is a basis for the vector space

There is a linearly independent set of vectors in the vector space $V$, given by $\{v_1, v_2,...v_k\}$. We have to show that the set $\{v_1, v_2,...v_{k-1}\}$ cannot be a basis for V. It is clear ...
1
vote
1answer
36 views

Why doesn't a linearly independent set of image vectors imply an injection?

While researching a question I had, I came across this post. Without reading the answers, I started working on it myself and eventually came up with a proof for this statement: Let $f \in ...
2
votes
2answers
36 views

Proof that matrix $B^{-1}$ = matrix $A^{-1}$ with 2 columns swapped given that B = A with 2 rows swapped.

I'm trying to prove the following. Given that $A$ is a nonsingular $n \times n$ matrix, and $B$ is the nonsingular matrix obtained by interchanging rows $i$ and $j$ of $A$, where $i \neq j$, show ...
0
votes
1answer
26 views

Images of basis vectors under injective linear map form a linearly independent set

I missed this question on a quiz: Prove that if $\{v_1, \ldots v_n \}$ is a basis for $V$ and $f\,:\,V\rightarrow W$ is an injective linear map, then $\{f(v_1), \ldots f(v_n)\}$ is linearly ...
1
vote
1answer
28 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 ...
1
vote
1answer
47 views

Orthogonal matrices, their determinant and eigenvalues

Once again! Let What to do? Find eigenvalues for $A, B, AB, BA$. How I want to do this: $A = \begin{pmatrix} cos(a) & -sin(a) & 0 \\ sin(a) & cos(a) & 0 \\ 0 & 0 & 1 ...
2
votes
3answers
53 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 = ...
3
votes
1answer
47 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
2answers
63 views

Let $(V,\vert\vert\cdot\vert\vert)$ be a Banach space. Prove if $W\subset V$ and $\dim(W)=n$ then $W$ is a closed subset of $V.$

The original (with words) problem statement is: Let $(V,\vert\vert\cdot\vert\vert)$ be a Banach space. If $W$ is a finite dimensional subspace of $V$ then $W$ is a closed subset of ...
0
votes
0answers
23 views

Understanding 2nd half rank-nullity theorem proof.

I'm trying to understand the second half of the rank-nullity theorem (the part that shows $T(e_{k+1}) \dots T(e_{k+r})$ is independent). Assume $e_1 ,\dots e_k, e_{k+1}, \dots e_{k+r}$,is a basis for ...
1
vote
0answers
39 views

Proof by contradiction: $E_1+E_2\doteq E_1 \oplus E_2 \leftrightarrow E_1 \cap E_2=\{0_V\}$

I must proof the following: Prop.: Let $E_1,E_2$ two vector subspace of $V$ then $$E_1+E_2\doteq E_1 \oplus E_2 \leftrightarrow E_1 \cap E_2=\{0_V\}$$ Proof: I must show $$1)E_1+E_2\doteq E_1 \oplus ...
2
votes
0answers
40 views

Show that $T = cL$ for some real number $c.$

Let $L$ and $T$ be two linear transformations from a real vector space $V$ to $\mathbb R$ such that $L(v) = 0$ implies $T (v) = 0.$ Show that $T = cL$ for some real number $c.$ I have taken hours to ...
1
vote
1answer
74 views

How does this proof of the Cauchy-Schwarz Inequality work?

I was watching a Khan Academy video on the Cauchy-Schwarz Inequality, and I just can't seem to understand the proof, and the comments on the video don't seem to help. The video is here. First, he ...
1
vote
1answer
28 views

Proof that we get equivalent system by applying elementary row operations

Let $A$ be system of liner equations. And let $A\leadsto A'$ mean that we got $A'$ by applying elementary row operations to $A$. Let $p_{ij}$ represent row switching, $q_{ij}(K)$ represent row ...
1
vote
2answers
115 views

Showing that $A=B+\alpha \cdot I$ is an invertible matrix

Let $B$ be a non-zero random $n\times n$ matrix generated using the matlab command $B=rand(n,n)$. I need to show that $A=B+\alpha \cdot I$ is an invertible matrix, where $\alpha=\|B\|_{\infty}$. I ...
8
votes
1answer
112 views

Validity of my weird proof that $AB$ and $BA$ have the same eigenvalues?

On a recent linear algebra exam, I was required to prove that "for every $n \times k$ matrix $A$ and $k \times n$ matrix $B$ over the same field, it holds that $AB$ and $BA$ have the same eigenvalues ...
1
vote
2answers
68 views

Correct proof? Linear Algebra

Prove that if $A$ and $B$ are matrices of rank $n$, then $AB$ is of rank $n$. Solution This should be equivalent to proving that the columns $AB$ are linearly independent. $AB = \begin{pmatrix} ...
2
votes
0answers
56 views

Does $Z_A$ exist such that $\exp(X+A) = \exp(X) Z_A$?

I am considering an exponential on the following form: $$\exp(X + A \otimes I_B),$$ where $X$ is a Hermitian operator on a tensor Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$, $A$ is a ...
0
votes
1answer
136 views

Proving equivalence relations

I just started my abstract algebra class and I am struggling with the concept of equivalence relations. I know that in order to prove equivalence relations I have to prove the reflexive, symmetric, ...
1
vote
2answers
50 views

Wronskian determinant and Linear dependence

I was trying to show that if functions f and g defined on interval I are linearly dependent then the Wronskian determinant is zero. Suppose f, g $\in$ I and f g are linearly dependent, then $\forall ...
2
votes
2answers
87 views

Are the proofs I made correct?

Edit: Since these are pretty small assignments each and all of the same topic, I've decided to post them into one thread. I hope that's ok. Thank you. Question I have the following assignment: ...
2
votes
1answer
93 views

Problem with alternate solution — Equation of plane through point and containing intersection line of two planes [Stewart P $803, 12.5.37$]

$37.$ Find an equation of the plane that passes through the point $(1, -2, 1)$ and contains the line of intersection of the planes $x + y - z = 2$ and $2x - y + 3z = 1$. $\bbox[3px,border:2px solid ...
1
vote
0answers
36 views

How to prove that the inverse of a persymmetric matrix is also persymmetric?

An exercise in a textbook I'm using to brush up on my linear algebra asks to prove that the inverse of a persymmetric matrix is also persymmetric. I have a colleague's old notes in front of me with a ...
5
votes
1answer
55 views

Prove that $n$ is even and $|A| \in \{-1,1 \}$

Let $A \in M_{n} (\mathbb R)$, such that $A^2=-I_{n}$. Prove that $n$ is even and $|A| \in \{-1,1 \}$. I started by compute the determinant of both sides: $A^2=-I_{n}\Leftrightarrow$ ...
0
votes
1answer
57 views

Proof for cosine formula for the dot product

I'm trying to understand how the geometric and algebraic formulas for the dot product are in equal. In doing so, I first am going through the proof that $V \cdot W = ||V||||W|| \cos{\theta}$ Here's ...
2
votes
0answers
42 views

Proof regarding the direct sum of subspaces

I was wondering if someone could verify that this proof is correct. I am also open to any critique. Thanks! Theorem: Suppose that $V$ is finite dimensional with $\dim{V}=n$. Prove that there exist ...
3
votes
0answers
44 views

Find signature of symmetric block matrix, given the diagonal blocks are positive / negative definite - Check my proof

This may be a basic question, but I'd like someone to double check it. We are given the matrix $A=\begin{pmatrix} A_1 & C \\ C^T & A_2\end{pmatrix}$ where $A_1$ is a $k$ by $k$ positive ...
0
votes
1answer
51 views

uniqueness of map proof

Let $W_1,W_2 \in V$ be complementary subspaces. Show that there exists a unique map $P$ with with Kern$P=W_2$ and Im$P=W_1$ such that $P=P^2$ I prooved the existence of such a map and it is in ...
1
vote
1answer
82 views

Proving summation identities [duplicate]

How would one go about proving the following identities? $$\sum_{i=1}^n \sum_{i\neq j}^n \frac{z_i}{z_i-z_j} = \frac{n(n-1)}{2}$$ $$\sum_{i=1}^n \sum_{i\neq j}^n \frac{z_i^2}{z_i-z_j} = ...
0
votes
0answers
38 views

Characterize graph by its connectivity matrix

Let $A$ be an $n\times n$ symmetric matrix, all of whose entries are $1$ or zero. Such a matrix is associated with an undirected graph $G$ with $n$ nodes, in which there is an edge between ...
3
votes
2answers
50 views

Show a matrix is normal - check my proof

Short easy question, I just want someone to double check what I did. We are given that $T$ is an invertible, normal matrix. We are asked to show that $T^{-1}$ is also normal, and find it's unitary ...
2
votes
1answer
78 views

Are $A,B$ similar matrices? Check my proof

We are given $A,B$ are orthogonal $4$ by $4$ matrices with real values only. We are given $\det(A) = \det(B) = 1$ and $\mathrm{trace}(A) = \mathrm{trace}(B)$. Is $A$ similar to $B$? My solution: I ...
3
votes
0answers
74 views

Question about orthogonal transformation / orthogonal matrices

I have a question about orthogonal transformations. If $T$ is an orthogonal transformation from $V$ to $V$, should the representation matrix with respect to any orthonormal basis of any inner product ...
1
vote
1answer
109 views

Can you check my proof on the characterization of the trace function?

The following is my proof: Theorem: If $W$ is the space of $n \times n$ matrices over the field $F$ and if $f$ is a linear functional on $W$ such that $f(AB)=f(BA)$ for each $A$ and $B$ in $W$, then ...
0
votes
1answer
53 views

Linear algebra hw! Linear transformation

Let $T : V -> V$ be a linear transformation where V is a nite dimensional vector space. If rank(T) = rank$(T^2)$, prove that image(T)$\cap$Ker(T) = {0}. I have to give this hw to my prof this ...
1
vote
1answer
41 views

proving $W_1 + W_2$ is a subspace of V

I'm just trying to clarify one of my answers here. Here is the question: if $W_1$ and $W_2$ are subspaces of a vector Space $V$, show that $W_1 + W_2 = \{x+y : x \in W_1, y \in W_2\}$ is a ...
2
votes
0answers
47 views

Show $r(F)=r(F^2)$ implies $Im(F) \cap Ker(F)=\{0\}$

I wonder if I've made some mistakes in the proof of the following or if there is some simpler solution. Problem: Let $V$ be a finite dimensional vectorspace and $F:V \rightarrow V$ a linear operator. ...
2
votes
1answer
64 views

If two matrices have the same 4 fundamental subspaces, then free variable submatrices must be equal. [GStrang P194 3.6.42]

$\Large{{1.}}$ Would someone please divulge and expound the intermediate steps? Here's my attempt. Suppose both matrices have rank $r$, WLOG. Then $I$ has size $r \times r$, $F$ and $G$ sizes $r ...
3
votes
1answer
76 views

Prove: Two vector spaces of equal dimension are isomorphic.

We were assigned the following problem for my linear algebra course and I was just wondering if somebody could critique/validate my proof. Let $U$ and $V$ be finite dimensional vector spaces such ...
1
vote
1answer
23 views

Help Finishing/Checking Infinity Basis

I would greatly appreciate if somebody could check over a proof I have written for a homework assignment. Write now, I feel that some things could be simplified a but, but I'm not quite sure how to do ...
2
votes
1answer
62 views

Using matrix theory to solve this problem

I'm sorry that I couldn't find a better title for this. I was wondering if my solution is valid for the following problem, or if I've made some mistake. Problem: Let $N=\{a_1, \dots, a_n\}$ be a ...
1
vote
2answers
193 views

Transpose of an invertible linear transformation..

I am trying to prove that suppose that a linear transformation $T$ is invertible, then its transpose $T^t$ is also invertible. Is the following proof correct? Proof: Let $T$ be an invertible ...
3
votes
2answers
142 views

Find a basis for the subspace sum and then calculate its dimension.

By definition, $U + V = \{\mathbf{u} + \mathbf{v} : \mathbf{u} \in U\ \; \& \; \mathbf{v} \in V\}$. Let $U = \{ \; u_1 = (1, 1, 0, \color{green}0), u_2 = (-3, 7, 2, \color{green}1) \;\}, V = ...