# How many vectors from the matrix are linearly independent?

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

• I believe the answer is 2 in both cases since the rank is 2. Is this correct? – zeeks Feb 26 '18 at 3:44
• or for $R^3$, there are 0 linearly independent vectors since there are 4 vectors and we have R^3? – zeeks Feb 26 '18 at 3:46
• Correct!${}{}{}$ – FYY Feb 26 '18 at 3:49
• Hint: the middle row is the sum of the other two. – dxiv Feb 26 '18 at 3:51

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.
• among the notes the professor gave us, he said if the number of vectors is higher than n in $R^n$, which in the first case is $R^3$, then the vectors are linearly dependent. does that mean, for the first question, the answer is 0? or i misunderstood his point. – zeeks Feb 26 '18 at 3:51
• No, it is still $2$. The answer is $2$ in the sense that out of all these vectors, you can find two of them such that they form a linearly independent set. Linearly dependent doesn't mean we can't find any such vector at all, the answer is not zero unless you have the zero matrix. – Siong Thye Goh Feb 26 '18 at 3:59