Math310: Exam 2
Problem 1. Let Sbe the subspace of R4spanned by
w1=
1
0
2
1
, w2=
1
1
2
0
, w3=
2
1
4
1
, w4=
3
2
6
1
.
a) (15 pts) Find a basis of S. What is the dimension of S?
b) (15 pts) Find a basis of S⊥. What is the dimension of S⊥?
Problem 2. Let u1=2
1, u2=3
2and v1=1
0, v2=−1
1.
a) (15 pts) Find the transition matrix Scorresponding to change of basis from {u1, u2}
to {v1, v2}.
b) (10 pts) If L:R2→R2is a linear operator such that L(v1) = v1+v2and L(v2) = v2,
find the matrix representation of Lwith respect to the basis {v1, v2}.
c) (10 pts) Compute L(u1).
Problem 3. (15 pts) Find the best possible straight line to fit the 3 data points (0,1),
(1,2) and (2,4).
Problem 4. Consider vector space C[−1,1] with inner product defined by
< f, g >=1
2Z1
−1
fgdx.
Let S= Span{1, x, x2+x+ 1}.
(20 pts) Use Gram-Schmidt orthogonalization process to find an orthonormal basis of S.