The column space of each matrix A and B below determines a plane in R 3
. We will find a non-zero vector in the intersection of the two planes, i.e., a vector v in C(A)∩C(B). A= ⎣
⎡
1
1
1
2
3
2
⎦
⎤
B= ⎣
⎡
5
6
5
4
3
1
⎦
⎤
We want to find v so that v=Ax and v=By for some x and y. (a) Let x=[ x 1
x 2
] and y=[ y 1
y 2
]. Rewrite the equation Ax=By as a 3×4 matrix system. (b) Find the complete solution to the system in part (a). Note that this should be the null space of the combined matrix [A∣−B]. (c) Take a non-zero solution to the system from part (b) and split it into an x-part and a y-part. This should give you vectors x and y with Ax=By. (d) Find a non-zero vector v in C(A)∩C(B).