I've been staring at these two problems for hours and I can't seem to figure them out. They seem to have nothing to do with the rest of the problem set and my professor said nothing about them in class... Will give a million thanks a kiss in advance!

1. Let

[-7 1 1 -7]

Find a basis of the subspace of

consisting of all vectors perpendicular to

.

[_ _ _ _ ], [_ _ _ _ ], [_ _ _ _ ]

You have to find a basis for the solution space of $\displaystyle -7x_1+x_2+x_3-7x_4=0$ (why?). This basis is what you want.
2.

Let

be the basis of

consisting of the vectors

[5 -3] and [2 5] ,

and let

be the basis consisting of

[-2 1] and [3 -2 ] .

Find a matrix

such that

for all

in

.

[_ _] <-----(2x2 matrix)

[_ _]

You have to find the matrix of the transformation that maps one basis into the other one (transition map/matrix)... Tonio
(All vectors should be column vectors, I had to change them because they didn't copy right)

Thanks in advance!