0
votes
0answers
41 views

Why does distance lose meaning in high-dimensional space?

I'm working on an algorithm that clusters points in extremely high-dimensional space (thousands, if not more). However, I came across this wikipedia page: ...
0
votes
3answers
41 views

Finding ker, im, dim of a linear transformation

1Ok, I am a student trying to wrap my head around some of these concepts and need help understanding how to approach some problems. Question: Let $\alpha:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ be the ...
1
vote
1answer
71 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
3k 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 = ...
3
votes
1answer
171 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
201 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}) = ...