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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: Precise Description, Row Operation, Kinds, Row Echelon Form, Matrix, Permitted, Elementary Matrix, Set of Solutions, System, Subspace
Typology: Exams
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December 15, 1999, 3:30 – 6:30 p.m.
Name: family name given name
Number:
Question Score Max
system
2 x 1 − x 2 − 6 x 3 + 10 x 4 = 0 −x 1 + 3 x 2 + 8 x 3 − 15 x 4 = 0.
ANSWER
a 1 = [2, 2 , 1 , − 1 , 0] a 2 = [− 1 , 1 , 1 , 2 , 2] a 3 = [7, 1 , − 1 , − 8 , −6] a 4 = [0, 8 , 6 , 6 , 8] a 5 = [1, 1 , 1 , − 1 , −3]
and A =
a 1 a 2 a 3 a 4 a 5
be the 5 × 5 matrix whose columns are a 1 , a 2 , a 3 , a 4 , a 5.
By elementary row operations A is converted to
[2] (a) Write down a basis for V. ANSWER
[2] (b) Write down a basis for the row space of A.
ANSWER
[2] (c) Determine the rank of A. Give a reason for your an- swer.
ANSWER
(use the back of page 2)
intersection W 1 ∩ W 2 is a subspace of V.
multiplication the usual addition of matrices and multiplication of a matrix by a scalar.
Given are four matrices
v 1 =
, v 2 =
, v 3 =
, v 4 =
It is given that B = (v 1 , v 2 , v 3 , v 4 ) is an ordered basis for R^2 ×^2.
Let
v =
Find the coordinate vector vB of v relative to B. ANSWER
(Question 7. continues here.)
[3] (b) Decide whether the transformation T 1 is invertible. Justify your answer.
[2] (c) Use the composition of linear transformations to discover a simple relation between A 2 and A 1. Justify your answer. Do not compute A 2 explicitly, just express it in terms of A 1.
Find the area of the triangle P QR.
ANSWER
[4] (b) State the row-interchange property for determinants of square matrices. Use it to prove:
If two rows of a square matrix A are equal, then det(A) = 0.
[5] (a) Find the eigenvalues and corresponding eigenspaces of A.
ANSWER
(Question 10. continues here.)
[3] (b) Use diagonalization to compute A^2000. Give your answer in the form of a single 3 × 3 matrix.
ANSWER
[3] (c) Decide whether the matrix
is diagonalizable. Justify your answer.
ANSWER
and a 3 = [0, 1 , − 1 , 0].
[6] (a) Find an orthogonal basis for V.
ANSWER
[2] (b) Use your answer to part (a) to find an or- thonormal basis for V.
ANSWER
SHOW YOUR WORK (use the back of the previous page if necessary)
Find the least-squares linear fit for these data points.