I am quite a newbie when it comes to math and I'm now studying linear algebra.
I have a assignment, where I have to find the column space C(A) and nullspace N(A) for a matrix A.
\begin{bmatrix}a&b&c\\d&e&f\end{bmatrix}
I understand what the terms mean but I don't understand how should I proceed in finding the C(A) and N(A) for a given matrix.
For example, let Matrix A be the following matrix:
\begin{bmatrix}0&0&3\\1&2&3\end{bmatrix}
What is the process of finding C(A) and N(A)? Thank you!
Best Answer
Finding the nullspace is in my opinion easier, because it amounts to solving an equation. More precisely, since the nullspace is the set of all vectors $x$ that the operation represented by matrix $A$ sends to the vector $0$, then you have the equation :
$$ x \in N(A) \Leftrightarrow Ax = 0 $$
Here your example is pretty simple because if you write this, with $x = \begin{pmatrix}a\\b\\c\end{pmatrix}$ :
$$ Ax = 0 \\ \Rightarrow \begin{cases} 3c = 0 \\ a + 2b + 3c = 0 \end{cases} $$
You immediately see that the only solution to this is $a = -2b, c=0$, that is, all vectors multiple of $\begin{pmatrix}-2\\1\\0\end{pmatrix}$.
The column space, however, is a bit trickier. However, since it is the set of all vectors that have an antecedent by $A$, you could also write an equation on $y = \begin{pmatrix}u\\v\end{pmatrix}$:
$$ y \in C(A) \Leftrightarrow \exists x, Ax = y \\ \Rightarrow \begin{cases} u = 3c \\ v = a + 2b + 3c \end{cases}\\ \Rightarrow v = a + 2b + u $$
You can then see that for any vector in $\mathbb{R}^2$ you can find a corresponding vector $x$, which means that the columnspace of $A$ is $\mathbb{R}^2$.
The only thing to be very careful about is the dimensions of your vector : the null space is in $\mathbb{R}^3$ because your matrix has 3 columns, the columnspace is in $\mathbb{R}^2$ because your matrix has 2 rows.