Range of matrix Set builder notation

Question

I am unable to to understand this. $$ \text{Range}(A) = \{v ∈ R^5: v = Ax \text{ for some }x\} $$ $$ \text{Range}(A^T) = \{v ∈ R^5: v = A^Tx \text{ for some } x\}** $$ Can anyone please explain this in simple words. this is related to matrix in Linear Algebra. also please explain what is $\Bbb R^5$. I know $\Bbb R$ is for real number what is 5 for is this telling us that it has 5 rows?

Answer 1

First of all, $\mathbb{R}^5$ is the space of vectors of length $5$ which can be thought of as "five-dimensional" Euclidean space. $\mathbb{R}^5$ can be written as $$ \mathbb{R}^5=\left\{\begin{bmatrix} a\\ b\\ c\\ d\\ e \end{bmatrix} :a,b,c,d,e\in \mathbb{R}\right\}.$$ One thing we know from basic linear algebra is that an $m\times n$ matrix $A$ can be viewed as a function $A:\mathbb{R}^n\to\mathbb{R}^m$ given by $\vec{x}\mapsto A\vec{x}$ for $\vec{x}\in \mathbb{R}^n$. In this case, I suppose that $A$ is a $5\times 5$ matrix, so that $A^T$ is also $5\times 5$. According to the definition, the range of a matrix $A$ is the set of vectors $\vec{v}$ in the target space which are "hit" by the transformation. So, to say $\vec{v}\in \text{Range}(A)$ means that we can find $\vec{x}\in \mathbb{R}^5$ such that $A\vec{x}=\vec{v}$. The same applies for $A^T$, since it also represents a matrix transformation.