- 2 -
Answer at least FIVE questions.
1. Consider the following basis of R3.
B=
1 1 1
,
2 1 1
,
2 2 1
(a) Find change of basis matrices 1U B and 1BU fromB to U, and from U to B.
(b) Write the matrix M =
1 1 0 1 0 1 0 1 1
in terms of basisB.
(c) Show that the following set of vectors does not form a basis.
S =
1 1 3
,
7
−1 7
,
11
−5 5
2. (a) Find an orthogonal matrixP and a diagonal matrixD so thatPtAP =D where
A=
1 2 0
2 0 2
0 2 −1
Note that|A−λI|=−λ3+ 9λ.
(b) A matrixM is calledidempotentifM2 =M. Show that the only possible eigenvalues for an idempotent matrix are 0 and 1.
(CONTINUED)
- 3 -
3. (a) Use the Gram-Schmidt method to find an orthonormal basis for
Span
1 0 0
−1
,
−1 1 0 5
,
3
−3 1 5
(b) IfA andB are Hermitian matrices, show that the eigenvalues ofA+B are real. State any theorem you use.
4. (a) Consider the matrix
A=
1 3 2
2 4 1
3 5 0
1 1 −1
Find invertible matricesP and Q so thatQ−1AP =
I 0 0 0
. You need not findQ−1.
(b) Find the adjugate of the following matrix.
A=
1 2 3 3 2 1 1 1 1
(c) Explain how the adjugate is related to the inverse. Hence show that for invertible matrices adj(AB) = adj(B)adj(A).
(TURN OVER)
- 4 -
5. (a) Write the following matrix as a product of elementary matrices
A=
2 9 1 2
(b) Find the eigenvalues of the matrix in part (a). You need not find the eigenvectors.
(c) List the axioms of a real inner product space, and use them to show that if hu+v, u−vi= 0 then hu, ui=hv, vi.
6. (a) Find the determinant of the following matrix
M =
2 3 2 3 1 2 1 2 1 2 2 1 2 3 3 2
(b) LetV be the inner product space of quadratic polynomials with real coeffi- cients under the operations of addition and scalar multiplication and with inner product defined by
hp, qi= Z 1
0
p(x)q(x)dx
This vector space has basis{1, x, x2}. Show that this basis isnotorthonor- mal, and use the Gram-Schmidt method to find an orthonormal basis of the inner product spaceV.
