Is vector space $\mathbb{R}^{n}$ the same that $\mathfrak{M}_{n\times 1}$?

Question

Being $\mathfrak{M}_{n\times 1}$ a vector space of matrix of $n$ rows and $1$ columns.

I can't found a substantial difference between $\mathfrak{M}_{n\times 1}$ and $\mathbb{R}^{n}$. Both have the same properties and I saw in Linear Algebra's books that are used interchangeably.

Answer 1

Yes . The map $$(x_1,x_2,\cdots x_n) \longmapsto \begin{pmatrix}x_1 &x_2& \cdots&x_n \end{pmatrix}^T \in M_{n \times 1}$$ is an isomorphism. So we can identify $M_{n \times 1}$ by $\Bbb R^n$. Consequently, $M_{n \times n}$ can be identified with $\Bbb R^{n^2}$