# Linear Algebra Review

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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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