# Isomorphisms between elements in $\mathbb{R}^n$ and $n \times 1$ column matrices

In the book Analysis on Manifolds by Munkres there is the following passage

Now in my exposure to Linear Algebra, I always thought that elements in $\mathbb{R}^n$ could be "represented" (informally) as $1 \times n$ row vectors (or row matrices in Munkres' terminology) or as $n \times 1$ column vectors (or column matrices in Munkres' terminology).

However Munkres states that there exists a linear isomorphism (vector space isomorphism) between elements of $\mathbb{R}^n$ and their corresponding row/column matrices. But while $\mathbb{R}^n$ is certainly a vector space, what is the other vector space(s) that Munkres is asserting that $\mathbb{R}^n$ is isomorphic to (as vector spaces)?

I'm guessing that it's the set of all $n \times 1$ matrices with entries in $\mathbb{R}$ and similarly the $1 \times n$ matrices with entries in $\mathbb{R}$.

• Yes, it's exactly that. – Bernard Jun 24 '18 at 19:49

$M_{m\times n}(\mathbb{R})$, the set of all $m\times n$ matrices with entries in $\mathbb{R}$ is a vector space (over $\mathbb{R}$). Mukners claims that $$\mathbb{R}^n \cong M_{1\times n}(\mathbb{R}) \cong M_{n\times 1}(\mathbb{R})$$
As you know every n-dimensional vector space over $\mathbb {R}$ is isomorphic to $\mathbb {R^n}$
$$\mathbb{R}^n \cong M_{1\times n}(\mathbb{R}) \cong M_{n\times 1}(\mathbb{R})$$ is a special case of above statement.
Another example is the set of $n\times n$ real diagonal matrices.