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MAT3701 LINEAR ALGEBRA EXAM REVISION MATERIAL
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MAT3701 LINEAR ALGEBRA EXAM REVISION MATERIAL. IT ALSO ASSISTS WITH THE WRITING OF ASSIGNMENTS.
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MAT3701May/June 2011
LINEAR ALGEBRA
Duration : 2 Hours 100 Marks
EXAMINERS :
FIRST : PROF JD BOTHA
SECOND : PROF TA DUBE
EXTERNAL : PROF LM PRETORIUS (PRETORIA - UP)
This paper consists of 3 pages
ANSWER ALL THE QUESTIONS.
QUESTION 1
Let V be the vector space C 2 with scalar multiplication over the real numbers R. It is given that
β = {(1, 0) , (i, 0) , (0, 1) , (0, i)}
is a basis for V. The mapping T : V → V is defined by
T (z1 , z2 ) = (z1 − z 1 , z2 + z 2 ) ,
where z1 and z2 are complex numbers and z denotes the complex conjugate of z.
(a) Show that T is a linear operator. (6)
(b) Find a basis for N (T ) . (6)
(c) Find a basis for R (T ) . (6)
(d) Determine whether V = N (T ) ⊕ R (T ) . (2)
[20]
QUESTION 2
Let T : P2 (C) → P2 (C) be the linear operator defined by
T a + bx + cx2 = (a + c) + (b + c) x + 2cx2 .
(a) Show that T satisfies the test for diagonalisability. (10)
(b) Find a basis τ for P2 (C) consisting of eigenvectors of T, and write down [T ]τ . (7)
(c) Determine whether T − IP2 (C) is a projection. (3)
[20]
2
, MAT3701/102
QUESTION 3
Let
1
0 0
2
1 1
A= .
1
2 2
1
0 0
2
(a) Show that T is a regular transition matrix. (4)
(b) Find lim Am . (8)
m→∞
(c) Describe the Gerschgorin discs in which the eigenvalues of A lie. (4)
[16]
QUESTION 4
Let P denote the orthogonal projection of C 3 on
1 1
W = span √ (1, 0, i) , √ (i, 0, 1)
2 2
(a) Find the formula for P (z1 , z2 , z3 ) . (7)
(b) Find the eigenvalues and corresponding eigenspaces of P. (4)
(c) Find the vector in W closest to (1, 1, 1) ∈ C 3 . (2)
[13]
QUESTION 5
It is given that A ∈ M3×3 (C) is a self-adjoint matrix with eigenvalues 1 and 2, and corresponding
eigenspaces
1
E1 = span (0, 1, 0) , √ (i, 0, 1)
2
1
E2 = span √ (1, 0, i)
2
Find the spectral decomposition of A.
[15]
3
, QUESTION 6
Let
−1 3
A= .
3 −1
(a) Find kAk , kA−1 k and cond (A) . (9)
(b) Suppose x and x e are vectors such that Ax = b, kbk = 1, and kb − Ae xk ≤ 0.001. Use (a)
to determine upper bounds for ke x − A−1 bk (the absolute error) and ke
x − A−1 bk / kA−1 bk (the
relative error). (7)
[16]
TOTAL: [100]
Memorandum: May/June 2011 Exam
Question 1
(a) T ((z1 , z2 ) + (z3 , z4 )) = T (z1 + z3 , z2 + z4 )
= (z1 + z3 − (z1 + z3 ) , z2 + z4 + (z2 + z4 ))
= (z1 + z3 − z 1 − z 3 , z2 + z4 + z 2 + z 4 )
= (z1 − z 1 , z2 + z 2 ) + (z3 − z 3 , z4 + z 4 )
= T (z1 , z2 ) + T (z3 , z4 )
T (a (z1 , z2 )) = T (az1 , az2 )
= (az1 − az 1 , az2 + az 2 )
= (az1 − az1 , az2 + az2 ) , a real
= a (z1 − z 1 , z2 + z 2 )
= aT (z1 , z2 )
Thus T is linear. (6)
(b) T (z1 , z2 ) = (z1 − z 1 , z2 + z 2 ) = 0 ⇔ z1 = z 1 , z2 = −z 2
∴ z1 = a real and z2 = ib imaginary
∴ (z1 , z2 ) = (a, ib) = a (1, 0) + b (0, i)
∴ N (T ) = span {(1, 0) , (0, i)} with basis α = {(1, 0) , (0, i)} (6)
(c) T (z1 , z2 ) = (z1 − z 1 , z2 + z 2 )
= (2ia, 2b) , a, b real
= 2a (i, 0) + 2b (0, 1)
∴ R (T ) = span {(i, 0) , (0, 1)} with basis β = {(i, 0) , (0, 1)} (6)
4
, MAT3701/102
(d) Since α ∪ β is a basis for V, it follows that V = N (T ) ⊕ R (T ) (2)
[20]
Question 2
(a) Let β = {1, x, x2 } .
T (1) = 1
T (x) = x
T (x2 ) = 1 + x + 2x2
1 0 1
∴ [T ]β = 0 1 1
0 0 2
∴ c (T ) = (x − 1)2 (x − 2)
0 0 1
[T ]β − I3 = 0 0 1 ,
0 0 1
hence rank [T ]β − I3 = 1 = 3 − (multiplicity of λ = 1)
∴ T is diagonalisable. (10)
(b) From (a),
E1 [T ]β = span {(1, 0, 0) , (0, 1, 0)}
∴ E1 (T ) = span {1, x}
−1 0 1
λ = 2 [T ]β − 2I3 = 0 −1 1
0 0 0
E2 [T ]β = span {(1, 1, 1)}
∴ E2 (T ) = span {1 + x + x2 }
1 0 0
Let τ = {1, x, 1 + x + x2 } , then Mτ (T ) = 0 1 0 (7)
0 0 2
(c) T − IP2 (C) is a projection since
0 0 0
[T − IP2 (C)]τ = 0 0 0 = B and B 2 = B. (3)
0 0 1
[20]
5
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