How many vectors from the matrix are linearly independent?

Question

I have the matrix:

$$ \begin{pmatrix} 3 & 2 & -1 & 4 \\ 1 & 0 & 2 & 3 \\ -2 & -2 & 3 & -1 \\ \end{pmatrix} $$

I have 2 questions to answer:

1. Consider the columns of the matrix as vectors in $R^3$. How many of these vectors are linearly independent?
2. Consider for $R^4$. How many vectors are linearly independent?

Is the answer in both cases 2 or am I totally wrong about how to solve this.

Thanks

Answer 1

I interpret the first part of the question as asking for the column rank of the matrix and the second part of the question as asking for the row rank of the matrix.

Yes, compute its RREF and you can see that there are two pivot columns, hence the rank is $2$.

It is known that the row rank is equal to the column rank and the answer is $2$ for both.