# dimension of column space and null space

I am trying to answer a series of questions given the dimension of a matrix.

Suppose that $$A$$ is a $$6 \times 12$$ matrix.

The column space is a subspace of $$\mathbb{R}^n$$. What is n?

$$n = 6$$ because there can only be 6 pivot columns.

The null space is a subspace of $$\mathbb{R}^m$$. What is m?

$$m = 12$$? Not so sure about this question.

Is it possible to have rank = 4, dimension of null space = 8?

$$rank \leq min(m,n)$$ for $$m \times n$$ matrix,

rank + nullity = number of columns.

It is possible.

Is it possible to have rank = 8, dimension of null space = 4?

rank + nullity = number of columns

but $$rank \nleq min(m,n)$$.

It is not possible.

Are my answers valid for the three questions that I answered?

I am a bit confused with the second question.

Any help would be great.

Thank you for reading.

The column space is a subspace of $$\mathbb{R}^n$$. What is n?

$$n = 6$$ because there can only be 6 pivot columns.

Your answer is technically correct, but misleading. I would say the following: the column-space is a subspace that contains the columns of $$A$$. Because the columns of $$A$$ each have $$6$$ entries, an element of the column space has $$6$$ entries which is to say that the column space is a subspace of $$\Bbb R^6$$.

The null space is a subspace of $$\mathbb{R}^m$$. What is m?

$$m = 12$$? Not so sure about this question.

Your answer is correct; here's a reason. The nullspace of $$A$$ is the set of column-vectors ($$k \times 1$$ vectors for some $$k$$) $$x$$ satisfying $$Ax = 0$$. However, in order for $$Ax$$ to make sense, the "inner dimensions" of $$m \times n, k \times 1$$ need to match, which is to say that $$k = n = 12$$. So indeed, the nullspace is a subspace of $$\Bbb R^{12}$$.

Is it possible to have rank = 4, dimension of null space = 8?

$$rank \leq min(m,n)$$ for $$m \times n$$ matrix,

rank + nullity = number of columns.

It is possible.

Is it possible to have rank = 8, dimension of null space = 4?

rank + nullity = number of columns

but $$rank \nleq min(m,n)$$.

It is not possible.

Both of these are correct; I have nothing to add.