Linear algebra - coordinates of vectors

Question

The question is: Compute the coordinates of vector c3, and vector c4 with respect to the basis of col(A).

The following is Matrix A: $$ \begin{bmatrix} 1 & 0 & 3 & 4\\ 2 & 1 & -1 & -2 \\ 3 & 1 & 2 & 2 \\ \end{bmatrix} $$

I was provided with RREF of matrix A, which is the following: $$ \begin{bmatrix} 1 & 0 & 3 & 4\\ 0 & 1 & -7 & -10 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} $$

I can easily see that the column space of A is: {[1 2 3], [0 1 1]}

Vector c1 should be referring to the first column of A; Vector c2 should be referring to the second column of A; Vector c3 should be referring to the third column of A; Vector c4 should be referring to the fourth column of A;

I was the told the answer should be computed by:

$$ \left[ \begin{array}{ccc|c} 1&0& &4\\ 0 & 1& &-10\\ 0&0& &0\\ \end{array} \right] $$

and

$$ \left[ \begin{array}{ccc|c} 1&0&& 3\\ 0&1&& -7\\ 0&0&& 0\\ \end{array} \right] $$

So the coordinates of vector c3 with respect to the basis of A is [4 -10 0] and the coordinates of vector v4 with respect to the basis of A is [3 -7 0]

This is a question I met in an assignment. I don't quite understand what it is asking. For the part of "basis of col(A)", why we're using the first and the second column vector in RREF, instead of that in matrix A? For the part of "vector c3" and "vector c4", why we're using the third and the fourth column of RREF, instead of that in matrix A?

If someone could explain what this question is expecting, it will be really helpful!

Answer 1

From the answers you were told, "basis of $A$" must mean "basis of the column space of $A$".

In the RREF of $A$, it's easy to see that $(\text{column }3) = 3\times(\text{column }1) - 7\times(\text{column }2)$. This relationship also holds in the original matrix $A$: $c_3 = 3c_1-7c_2$. The coordinates of $c_3$ with respect to the basis $(c_1,c_2)$ are the coefficients $\begin{bmatrix}3\\-7\end{bmatrix}$

Similarly we can read the relationship $c_4=4c_1-10c_2$ in both the RREF of $A$ and the original matrix $A$, and read off the coordinates of $c_4$ with respect to the basis $(c_1,c_2)$ as $\begin{bmatrix}4\\-10\end{bmatrix}$

So if someone gives you a matrix $A$ and asks you to find a basis of the column space of $A$, you could:

• compute the REF of $A$ (not necessarily the reduced REF)
• find the columns in the REF with leading $1$s. The matching columns in the original matrix $A$ are a basis for the column space of $A$

If you need to compute other columns of $A$ in terms of the basis you've found, then compute the full RREF and read the linear combinations straight from the RREF of $A$.

This method is a handy way to work out whether a bunch of vectors is linearly independent: put them in a matrix and compute the REF; any columns without leading $1$s are linear combinations of the columns with leading $1$s. If you need the exact linear combinations, it's easier to go to the RREF.