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This is the Exam of Applied Linear Algebra which includes Inconsistent, Matrix, Parallelogram, Precise Description etc. Its key important points are: Matrix Equation, Augmented Matrix, Linear System, Associated, Example, Solution, Row Echelon, Pivot Entry, Dimension, Row Echelon
Typology: Exams
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PLEASE PRINT (FAMILY NAME) (GIVEN NAME) (SFU ID)
SIGNATURE Simon Fraser University Department of Mathematics Final Examination MATH 232 12 December 2005 8:30am – 11:30am
Question Score Maximum
1 4
2 8
3 10
4 12
5 10
6 6
7 6
8 10
Total 66
[4] 1. Let A =
, b =
. Give the augmented matrix of the linear
system which results from the matrix equation AT^ Aˆx = AT^ b. We have that
AT^ A =
AT^ b =
The augmented matrix of the associated linear system is thus ( 1 1 0 1 1 0
. Suppose that b 1 =
(^) , b 2 =
(^) , b 3 =
(^) are eigenvectors for A.
[2] (a) What are the eigenvalues corresponding to each of b 1 , b 2 , b 3?
Hence the eigenvalues of b 1 , b 2 , b 3 are 1 , 2 , 2 respectively.
[4] (b) Is A diagonalizable? If so, give an invertible matrix P and a diagonal matrix D such that A = P DP −^1. A is diagonalizable as {b 1 , b 2 , b 3 } is a basis of R^3 consisting of eigenvectors of A. The matrix P and D are given by
[4] (c) Let λ 2 be the eigenvalue corresponding to the eigenvector b 2 of A. Find a basis for the eigenspace of A corresponding to the eigenvalue λ 2. Justify your answer carefully.
Note that λ 2 = 2 and both b 2 , b 3 ∈ EA(2) and are linearly-independent. We cannot have that EA(2) = R^3 as there is an eigenvector with eigenvalue 1 , namely b 1. Hence, dim EA(2) = 2 and {b 1 , b 2 } forms a basis for EA(2).
[4] 5. (a) Find the characteristic polynomial of A =
We have that
pA(λ) =
1 − λ 0 1 1 0 1 − λ 1 1 0 0 2 − λ 4 0 0 4 2 − λ
= (1 − λ)^2
2 − λ 4 4 2 − λ
= (1 − λ)^2 (4 − 4 λ + λ^2 − 16) = (λ − 1)^2 (λ^2 − 4 λ − 12) = (λ − 1)^2 (λ − 6)(λ + 2)
[4] (b) Determine a basis for the eigenspace of A corresponding to the eigen- value 1. We wish to determine a basis for Nul(A − I).
A basis is given by
[2] (c) Is A orthogonally diagonalizable? A is not orthogonally diagonalizable as A is not symmetric.
,^ b^2 =
,^ b^3 =
,^ b^4 =
[4] (a) Find an orthonormal set C such that Span {b 1 , b 2 , b 3 , b 4 } = Span C. We note that Span {b 1 , b 2 , b 3 , b 4 } = Span {b 1 , b 2 , b 4 }. We apply Gram- Schmidt to get an orthogonal basis first.
v 1 = b 1 =
v 2 = b 2 −
b 2 · v 1 v 1 · v 1
v 1
v 4 = b 4 −
b 4 · v 1 v 1 · v 1
v 1 −
b 4 · v 2 v 2 · v 2
v 2
Then
√^1 2 v^1 ,^ √^1 5 v^2 ,^ v^4
is an orthonormal basis for W.
[4] 7. (a) Suppose that A is a square matrix such that AT^ A = I. Prove that det A = ± 1. Since AT^ A = I, we have that det AT^ det A = 1 and hence (det A)^2 = 1 as det A = det AT^. Thus, det A = ± 1.
[2] (b) Let C be an orthonormal basis for R^3. What is the volume of the parallelpiped spanned by the vectors in C? Let A be the matrix whose columns are the vectors in C. Then AT^ A = I as C is orthonormal. The volume of the associated parallelepiped is |det A| = 1
[2] (a) What is the dimension of V? The dimension of V is 3.
[2] (b) What is the maximum number of elements in a linearly-independent subset of V? The maximum number of elements in a linearly-independent subset of V is 3.
[2] (c) Suppose T : V → V is a linear transformation such that T (b 1 ) = 2 b 1 + b 2 , T (b 2 ) = b 3 , T (b 3 ) = b 1 + b 3. Determine [T ]B. We have that
[T ]B =
[4] (d) Suppose T : P 1 → P 2 is the linear transformation given by T (a 0 + a 1 t) = a 0 t + a 1 t^2. Let B be the standard basis for P 1 and C be the standard basis for P 2. Determine [T ]C←B. We have that
[T ]B =
