0
votes
1answer
174 views

If $(V,k)$ is a finite-dimensional vector space, then the space of all linear transformations on $V$ is finite dim and find its dim?

My issue with this is the only way I know how to prove it is to set $\dim V=n$, but then that wouldn't make sense because the second part is find the $\dim$. What I was thinking is using the ...
2
votes
1answer
86 views

Radial coordinate evaluation

Details of the question can be found in the article equation(55,56) A radial coordinate $R$ defined by \begin{equation} r=\frac{2R}{\kappa(1-R^2)} \,, \end{equation} where $\kappa$ is a constantand ...
-1
votes
3answers
123 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 ...
4
votes
3answers
132 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 ...
0
votes
1answer
44 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
1answer
66 views

Dimension Recovery of $S \subset P_n(F)$

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset? Added from answer posted by Trancot on 18 Apr ...
-3
votes
4answers
117 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
68 views

Diagrammatic Representations: $\dim(Skew_{n\times n}(\mathbb{R}))+\dim(Sym_{n\times n}(\mathbb{R})) = \dim(M_{n\times n}(\mathbb{R}))$

SEE AUTHOR'S ANSWER BELOW So I'm trying to derive the dimensions of both $Skew_{n\times n}(\mathbb{R})$ and $Sym_{n\times n}(\mathbb{R})$. I know that $\dim(M_{n\times n}(\mathbb{R}))=n^2$, but I ...
2
votes
2answers
2k views

Dimension of the corresponding eigenspace?

I'm studying for my linear exam and would appreciate any help for this practise question: You are given that λ = 1 is an eigenvalue of A. What is the dimension of the corresponding eignspace? A = ...
1
vote
0answers
58 views

Buckingham Π-Theorem

I'm about to conduct some experiments concerning a welding process. To prepare for this I wanted to do a dimensional analysis of the process. So I read a lot about the Π-Theorem and I was able to ...
3
votes
1answer
166 views

Matrices to model 3D object

I'm toying around with an algorithm to determine placement of 3D objects into a larger 3D space. I immediately thought of using matrices. It's been some years since my Linear Algebra courses. I was ...
0
votes
1answer
66 views

Dimensions of Matrices Range (equalities).

I’d like to find range equalities. Considering the following: $$ A=B+C \\ A=B.C^T \\ A=[ B^T C^T ]^T \\ $$ I would like to find the function $f$ for each equality above. $$ dim( R(A) ) = f( R(B) , ...
2
votes
1answer
194 views

Linear Algebra Question ( rank of matrix )

Let $\bf A$ be an $m \times n$ matrix. If $\bf P$ and $\bf Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively prove $\operatorname{rank}(\mathbf{PA}) = ...
2
votes
5answers
2k views

Orthogonal planes in n-dimensions

In three dimensions two planes are orthogonal when their normal vectors are orthogonal (their inner product is zero). For example, planes $xy$ and $xz$ are orthogonal because the vectors $\hat{z}$ and ...