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Review
1 1
1 − 1
1. For the matrix
1 1
3 2
2
1 6
1 2
5 10
1
a) Find a basis for the null space.
b) Find a basis for the column space.
2. If
P2
is the vector space of polynomials in t with degree 2 or less and
T : P2 → R
2
p(0)
p(1)
is defined by T ( p ) =
then
a) Prove that T is a linear transformation.
b) Find a basis of kernel T
c) Describe the range of T ?
3. a) Show that
1 1
B = ,
2 1
is basis of
R2
b)
3 2
2
Given another basis C = , of R , write the change of
2 2
coordinates matrix
i)
ii)
from
from
B
C
to C
to B
1
1 1
4. Let A= 1 − 1 1
1 1 − 1
i)
ii)
Find the eigenvalues of A.
Diagonalize A, if possible.
Show all your calculations. Do not just write a calculator/computer output.
5. Find an orthonormal basis for the subspace of R 5 spanned by
5 3 1
− 3 − 1 1
0 , 2 , − 1
0 1 3
0 1 2
Show all your calculations. Do not just write a calculator/computer output.
6. Use the Grahm Schmidt process to find an orthogonal basis for the
subspace of C[−1,1] spanned by {1,1+x, 1 + x − 2x 2 }. Use
1
fgdx
as the
−1
inner product of f and g in C[−1,1].
7. Let A be a square nxn matrix and an eigen value of A. Let W=
v R n :Av = v . Prove that W is a subspace of R n .
8. Let W be a subspace of a vector space V with a specified inner product.
If W ⊥ contains the vectors of V that are orthogonal to W, prove that W ⊥
is a subspace of V.
9. Let T : V
spaces
V
V
and
→W
W
Prove that
and
be a one-one linear transformation between the vector
S = v1, v2 ,...., vn be a linearly independent subset of
T (v1 ), T (v2 ),...., T (vn ) is a linearly independent subset of
W
i) Write the quadratic form 5 x12 − 4 x1x2 + 5 x2 2 in terms of matrices.
ii)Find a coordinate transformation so that equation 5 x12 − 4 x1x2 + 5 x2 2 −21=0
transforms into an equation without the product term.
10.
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