Understanding relationship between row space and null space I was given the following question:

If the null space of a $7\times 5$ matrix $A$ is 4-dimensional, what is the dimension of the row space of $A$?

I know the answer to this question is $1$ - which is the number of columns minus the dimensions of the null space. Does this mean that the dimensions of the null space plus the dimensions of the row space equal the number of columns of the matrix? What is the row space?
Does this mean that the dimensions of the null space plus the dimensions of the row space equal the number of columns of the matrix?

No. This sum is equal the dimension of the domain of the linear transformation given by the matrix $A$. Since the domain is $\mathbb{R}^5$, it is equal tot 5.

What is the row space?

It is the vector subspace in the domain of a linear transformation that is spanned by the row vectors of the matrix $A$.
Does this mean that the dimensions of the null space plus the dimensions of the row space equal the number of columns of the matrix?

No. This sum is equal the dimension of the domain of the linear transformation given by the matrix $A$. Since the domain is $\mathbb{R}^5$, it is equal tot 5.

What is the row space?

It is the vector subspace in the domain of a linear transformation that is spanned by the row vectors of the matrix $A$.