Tagged Questions
1
vote
1answer
35 views
$(P\Lambda P^{-1}=T^2)~\implies~(\exists \Lambda'~\text{s.t.}~T=R\Lambda' R^{-1})$: $\;P,R\;$ Unitary Matrices
Let $T$ be a linear operator such that the operator $T^2$ is diagonalizable. Is $T$ necessarily diagonalizable?
3
votes
1answer
70 views
$V=W_1\oplus\cdots\oplus W_k \iff \dim(V)=\sum{\dim(W_i)}$
If $W_1,\dots, W_k$ are subspaces of a finite dimensional vector space $V$ such that $W_1+\cdots+W_k=V$, and I want to show that $V=W_1\oplus\cdots\oplus W_k$ if and only if $\dim(V)=\sum{W_i}$, then ...
0
votes
1answer
44 views
$V'=W_1\oplus (V'\cap W_2)$
Question:
Attempt:
Let $W_1$ be a subset of $V'$ spanned by the basis vectors $u_1,\dots,u_r$. Then $W_1$ can similarly be completed to a basis of $V'$, say $u_1,\dots,u_r,u_{r+1},\dots,u_s$. ...
-2
votes
3answers
65 views
A question about direct sums of subspaces…
Let $W_1$ be a subspace of a finite dimensional vector space $V$. Prove that there exists a subspace $W_2\subset V$ such that $V=W_1\oplus W_2$.
EDIT$^1$:
This may be of use here:
What ...
1
vote
3answers
43 views
0
votes
3answers
59 views
Diagonalizable Operators: An Operational Extension
Let $T$ be a diagonalizable operator on a vector space $V$. Prove that the operator
$$a_nT^n + a_{n-1}T^{n-1}+\cdots+a_1T+a_0 Id_V$$
on $V$ is also diagonalizable for any scalars $a_1, ...
2
votes
1answer
47 views
Basic concepts in finite fields
I need some help with clearing up some some basic concepts in finite fields.
I understand that $\mathbb{F}_p = GF(p)$ where $p$ is a prime is a finite field, which is isomorphic to ...
0
votes
3answers
29 views
$n$-linear alternating form with $\dim{V}<n$ $\overset{?}{\text{is}}$ the $0$-form
Prove that every $n$-linear alternating form on a vector space of dimension
less than $n$ is the zero form.
1
vote
0answers
30 views
A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$
Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$.
$F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$.
$P_1(F)$ is the set of all ...
0
votes
1answer
37 views
Bilinear Forms: An Initial Condition Proof
Let $B$ be a bilinear form on a finite dimensional vector space $V$. Suppose that for any nonzero vector $v \in V$ there exists a $w \in V$ such that $B(v, w)\neq 0$. Prove that for any linear ...
3
votes
1answer
71 views
$AX=C$: An Inconsistent Linear Equation [duplicate]
Question:
Let $A \in M_{n\times n}(F)$. Suppose that the system of linear equations $AX = B$
has more than one solution. Prove that there is a column $C \in F^n$ such that
the system of linear ...
1
vote
2answers
76 views
Inconsistent System of Linear Equations
Let $A ∈ M_{n\times n}(F)$. Suppose that the system of linear equations $AX = B$
has more than one solution. Prove that there is a column $C ∈ F^n$ such that
the system of linear equations $AX = C$ is ...
1
vote
2answers
64 views
Newbie vector spaces question
So browsing the tasks our prof gave us to test our skills before the June finals, I've encountered something like this:
"Prove that the kernel and image are subspaces of the space V: $\ker(f) < V, ...
3
votes
2answers
62 views
Silliness: $\exists~X~\text{s.t.}~AX=B \iff B\in R(L_A)$
So, I am asked to prove that the system of linear equations $AX=B$ has $\color{black}{a~solution}$ if and only if $B\in R(L_A)$. $R$ denotes the "range of" and $L_A$ is left multiplication by $A$. If ...
1
vote
1answer
32 views
$\operatorname{rank}(A\in M_{m\times n}(F)) =m \implies \exists~B\in M_{n\times m}(F)$ s.t. $AB=I_m$
Let $A ∈ M_{m×n}(F)$ be a matrix with $\operatorname{rank}(A) = m$. I just need some help showing that there exists a matrix $B ∈ M_{n×m}(F)$ such that $AB = I_m$.
2
votes
2answers
43 views
$AB = I_m \overset{?}{\implies} n\geq m$
Let $A ∈ M_{m×n}(F)$ and $B ∈ M_{n×m}(F)$ be two matrices such that $AB = I_m$. What should I be thinking to prove that $n ≥ m$?
4
votes
2answers
67 views
(sur/in)-jectivity
I'm having trouble showing this:
Let $T : V → W$ be a linear map of finite dimensional vector spaces. Prove that $T$ is surjective (respectively, injective) if and only if $T^*$ is injective ...
0
votes
1answer
52 views
What is a better way to state this?
Let $T : V → W$ be a linear map of finite dimensional vector spaces. Prove that $T$ is surjective (respectively, injective) if and only if $T^*$ is injective (respectively, surjective).
What is a ...
0
votes
0answers
76 views
Space-Function Cross-Element Interplay
Has anyone ever thought to show these results diagrammatically?
$$(W_1+W_2)^0 = W^0_1\cap W^0_2$$
$$(W_1\cap W_2)^0 = W^0_1+W^0_2$$
I mean with set nesting and overlap and containment and mappings ...
0
votes
1answer
64 views
$W^0$ is a subspace of $V^*$
If $W\subset V$ is a subspace of $V$, and $W^0=\{f\in V^* | f(v)=0~\forall~v\in W\}$, then how do I show that $W^0$ is a subspace of $V^*$?
0
votes
0answers
27 views
Normal endomorphism
I have a question about normal endomorphism. In class, we said that normal endomorphisms in finite dimensional real vector spaces are always of the form that we have some eigenvalues and further ...
0
votes
1answer
84 views
$(U\circ T)^{*} = T^{*}\circ U^{*}$
Let $T : V \longrightarrow W$ and $U : W \longrightarrow Z$ be linear maps. How do I prove that $(U\circ T)^{*} = T^{*}\circ U^{*}$? I'm used to seeing $V^{*}$ not $(U\circ T)^{*}$. Any help is ...
1
vote
2answers
30 views
$T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$
Question
Let $T:V \rightarrow V$ be a linear map. How do I prove that $T \circ T = T_0$ ( the zero linear map) iff $R(T) \subset N(T)$?
Attempt
\begin{eqnarray}
T\circ ...
-7
votes
0answers
124 views
Linear Algebra Midterm Review: Part II [closed]
Just as the first and the third, I've completed all of these problems below, but I'd like to see how other people do them too. Any help is greatly appreciated.
My "solutions" are below:
The ...
-2
votes
2answers
38 views
$T:V\rightarrow W$ such that $R(T) \subset W'$ is a subspace of ${\cal{L}}(V,W)$ [closed]
Let $V$ and $W$ be finite-dimensional vector spaces over $F$ and $W'\subset W$ a subspace, then the subset ${\cal{L}}(V,W)$ consisting of all linear maps $T:V\rightarrow W$ such that $R(T) \subset W'$ ...
0
votes
1answer
45 views
Existence Proof: $T(v_i)=w_i$ for all $i=1,2,3,\dots,n$
Theorem to prove:
Let $\{v_1,\dots,v_n\}$ be a linearly independent set in a finite-dimensional vector space $V$ and let $w_1,\dots,w_n$ be arbitrary vectors in a vector space $W$. Then there exists ...
2
votes
1answer
58 views
$[T]^{\beta}_{\beta} = \begin{pmatrix} I_k & 0 \\ 0 & 0 \end{pmatrix}$ provided $T \circ T = T$ [closed]
Let $V$ be a finite-dimensional vector space and let $T:V \rightarrow V$ be a linear map such that $T \circ T = T$. How should one prove that there is a basis $\beta$ of $V$ such that
\begin{eqnarray}
...
1
vote
3answers
68 views
Revisited: $[T]^{\gamma}_{\beta}$ is diagonal?
Let $V$ and $W$ be finite-dimensional vector spaces with $\dim(V)=\dim(W)$ and let $T:V \rightarrow W$ be a linear map. How do I prove that there are bases $\beta$ of $V$ and $\gamma$ of $W$ such that ...
0
votes
1answer
23 views
Equal Shape: Recovering an Isomorphism Between $M_{3\times 2}(F)$ and $P_5(F)$
I'm asked to find an isomorphism between $M_{3\times 2}(F)$ and $P_5(F)$, but what does it mean for a $3\times 2$ matrix to have an inverse?
4
votes
4answers
95 views
$T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$
How should one prove that there exists a linear map $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$ if $\dim(V')+\dim(W')=\dim(V)$, where $V$ and $W$ are finite-dimensional ...
2
votes
3answers
74 views
Revisited: How is $\phi:{\cal{L}}(V,W)\rightarrow M_{m\times n}(F)$ an isomophism of vector spaces?
I'm told in lecture that if $V,W$ are vector spaces over $F$ and ${\cal{L}}(V,W)$ is the vector space of all linear maps $V\rightarrow W$ and ${\scr{B}}$ and ${\scr{C}}$ are bases for $V$ and $W$ ...
0
votes
1answer
45 views
$T:P_4(\mathbb{R})\rightarrow P_4(\mathbb{R})$ such that $N(T) = P_1(\mathbb{R})$ and $R(T)=P_2(\mathbb{R})$
So, I'm asked to give an example of a linear map $T:P_4(\mathbb{R})\rightarrow P_4(\mathbb{R})$ such that $N(T) = P_1(\mathbb{R})$ and $R(T)=P_2(\mathbb{R})$.
As far as I understand, I'm trying to ...
-4
votes
0answers
127 views
$[T]_{\scr{C}}=[T]_{\scr{B}}^{\scr{C}}[v]_{\scr{B}}$: A Desideratum for Creative Disquisition [closed]
$\blacklozenge\hspace{.2cm}$Consider the following data in this first litany:
$\hspace{2cm}\Diamond\hspace{.5cm}$$V$ and $W$ are finite-dimensional vector spaces.
...
0
votes
1answer
30 views
Codimensionality: On Cardinality of Linear Equations
How does the codimension of a subspace give the number of linear equations needed to define the subspace?
1
vote
2answers
101 views
$\infty$ | Span | Basis
Let $V$ be a finite dimensional vector space and $S \subset V$ a subset (possibly
infinite) with $\operatorname{span}(S) = V$. Does there exist a subset of $S$ that is a basis for $V$?
-2
votes
4answers
100 views
Dimensionality and Subspace Existence: A Potential Outlet for Disquisition
The subset of $F^n$ consisting of all vectors $(a_1,a_2,\dots,a_n)$ such that $a_1+a_2+\cdots+a_n=0$ is a subspace of $F^n$ and its dimension is ...(?)....
Initially, my intuition said the ...
1
vote
1answer
34 views
On the Dimensionality of Space: An Elementary Analysis
The below theorem I am to prove. Perhaps you have a critique...
Theorem 2.4 Let $W_1$ and $W_2$ be two subspaces of a vector space $V$. Then $\dim(W_1 \cap W_2)=\dim(W_1)$ if and only if $W_1 ...
1
vote
1answer
66 views
Kenneth Hoffman | Ray Kunze: An Inquiry into Symbolic Meaning
When those authors state the following
$\bf{Theorem 6.}$ If $W_1$ and $W_2$ are finite-dimensional subspaces of a vector space $V$, then $W_1+W_2$ is finite-dimensional and
\begin{eqnarray}
\dim ...
1
vote
1answer
35 views
Orthogonal complement of S in a finite field $\mathbb F_q$
For the following set S and corresponding finite field $\mathbb F_q$, fing the $\mathbb F_q$-linear span $\left<S\right>$ and its orthogonal complement $\left<S\right>$-perp.
$$S = ...
0
votes
2answers
42 views
Counter Example: Span Inclusion
For any two subsets $S$ and $S'$ of a vector space $V$ does $span(S) \cap span(S') = span(S \cap S')$?
If $S=ax, a \in \mathbb{R}$ in $\mathbb{R}^2$ and $S'=by, b \in \mathbb{R}$ in $\mathbb{R}^2$ ...
2
votes
2answers
132 views
If a subset $S$ of a vector space $V$ is a subspace of $V$, then is $\langle S \rangle = S$?
I'm reading here on page 22 of Axler, Linear Algebra Done Right, where the following is stated:
A $\bf{linear}$ $\bf{combination}$ of a list $(v_1,\dots,v_m)$ of vectors in $V$ is a vector of the ...
0
votes
0answers
36 views
How to show that $\mathrm{Sym}_{n\times n}(\Bbb{R})$ and $\mathrm{Skew}_{n\times n}(\Bbb{R})$ are subspaces of $\mathrm{M}_{n\times n}(\Bbb{R})$
A matrix $M \in \mathrm{M}_{n\times n}(\mathbb{R})$ is called symmetric (respectively, skew-symmetric) if $M^t = M$ (respectively, $M^t = -M$). How does one prove that the sets $\mathrm{Sym}_{n\times ...
-6
votes
2answers
127 views
Intersection | Subspaces | Span
If $W_1$ and $W_2$ are two subspaces of a vector space $V$, then $W_1+W_2$ is the intersection of all subspaces of $V$ that contain $W_1$ and $W_2$, right? Is the intersection of all subspaces of $V$ ...
3
votes
1answer
57 views
Some questions about quaternions.
It is possible make something like complexification of a real vector space using quaternions?
If yes, it's similar to complex case or there are considerable differences?
Has been studied a quaternion ...
1
vote
1answer
62 views
Tensor algebra wedge
Let $k$ be a field and $V$ a vector space of dimension $n$. Let $P$ denote the image of the universal
alternating map $V \times V \to \bigwedge^2(V)$. (Thus $P$ consists of pure products of the form ...
1
vote
6answers
105 views
Proof that $\mathbb{R}[x]$ is not a finite dimensional vector space
How can we prove that the vector space of polynomials in one variable, $\mathbb{R}[x]$ is not finite dimensional?
2
votes
1answer
39 views
Definition of direct products of two cones or of two convex subsets?
When reading a comment after this reply, I was wondering what the definitions of direct product of two cones?
More generally, what is the direct product of two convex subsets? This case is what I ...
10
votes
1answer
100 views
Orientation on finite dimensional vector spaces over finite fields.
For finite-dimensional $\mathbb R$-vector spaces, we define an orientation to be an equivalence class of ordered bases, where $B_1 \sim B_2$ iff the change of basis matrix $A$ taking $B_{1}$ to ...
4
votes
1answer
103 views
Tensor product of real numbers over the rationals
How do I show that $\mathbb{R}\otimes_{\mathbb{Q}}\mathbb{R}\not\cong\mathbb{R}$ as $\mathbb{R}$-vector spaces?
Possible approaches I can think of (but can't implement) are to show that this tensor ...
2
votes
1answer
89 views
Questions about Hamel basis for ${\mathbb R}$ over ${\mathbb Q}$
If I understand the whole Hamel basis idea correctly, there exists one such basis $B = \{v_\alpha\}_{\alpha \in I}$ for ${\mathbb R}$ (herein construed as a vector space of ${\mathbb Q}$), such that ...
