$A$ is a vector subspace of $ℝ^4$? 
Am I right to say row space of the matrix $A$ is a vector subspace of $ℝ^4$? I think that because $\operatorname{Col}(A)$ is a vector subspace of $\mathbb{R}^4$ and $\dim \operatorname{Col}(A) = \dim \operatorname{Row}(A)$.
 A: The dimension of a vector space is the number of vectors in a basis. This doesn't say anything about what kind of vectors are in the vector space. You can have a one-dimensional subspace of $\mathbb R$, or $\mathbb R^2$, or of $\mathbb R^{100}$.
The row space is a subspace of $\mathbb R^5$, because its elements (linear combinations of the rows) are $5$-dimensional vectors. The column space is a subspace of $\mathbb R^4$, because its elements (linear combinations of the columns) are $4$-dimensional vectors.
For this matrix, the row space is a $2$-dimensional subspace of $\mathbb R^5$, and the column space is a $2$-dimensional subspace of $\mathbb R^4$, and there's nothing wrong with that.
A: Since the rows each have 5 entries, you can see that the row space must be a subspace of $\Bbb{R}^5.$ The column space is a subspace of $\Bbb{R}^4.$
You can generalize this by considering your matrix as a linear transformation, $A:\Bbb{R}^5 \longrightarrow \Bbb{R}^4$
Where $A: \text{domain}\longrightarrow \text{codomain},$ more generally, and so your row space will be a subspace of your domain, and your column space will be a subspace of your domain.
