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MAST10007 Linear Algebra Practice Class 10: Exercises and Solutions, Exercises of Mathematics

Mathematics 123 Mathematics Mathematics

Typology: Exercises

2018/2019

Uploaded on 10/06/2023

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MAST10007 Linear Algebra
Practice class 10
Q1. Let Bbe the basis B={(1,3),(2,1)}and let Sbe the standard basis of R2.
(a) Write down PS,B.
(b) Find PB,S.
Let u= (6,2).
(c) Find [u]Busing your answer from (b).
Let T:R2R2be the linear transformation given by
T(x, y)=(3x+ 2y, 3x+ 4y)
(d) Write down the standard matrix representation of T.
(e) Compute [T]B.
(f) Find [T(u)]B.
(g) Calculate [T(b)]Bfor each of the basis vectors b B. Hence describe the geometric effect
of the linear transformation T.
Q2. Let the change of basis matrix from B={b1,b2,b3}to C={c1,c2,c3}be
PC,B=
0 0 1
1 0 1
2 1 3
.
(a) Find the coordinate vector of v=b1+ 2b2+b3in basis C.
(b) Find the coordinate vectors of the vectors b1,b2,b3in basis C.
(c) Check that
0 0 1
1 0 1
2 1 3
1
=
1 1 0
52 1
1 0 0
.
Hence write down PB,C
(d) Using PB,C, compute the coordinate vector of c1+ 7c3in basis B. Check that your answer
is consistent with (a).
Q3. Let A=
3 0 1
0 2 0
2 0 0
(a) By calculating Av, verify that the vector v=
1
0
1
is an eigenvector of A. What is the
corresponding eigenvalue?
(b) Explain why u=
1
0
0
is not an eigenvector of A.
Mathematics and Statistics 1 University of Melbourne
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MAST10007 Linear Algebra

Practice class 10

Q1. Let B be the basis B = {(1, 3), (2, 1)} and let S be the standard basis of R^2. (a) Write down PS,B. (b) Find PB,S. Let u = (− 6 , 2). (c) Find [u]B using your answer from (b). Let T : R^2 → R^2 be the linear transformation given by T (x, y) = (− 3 x + 2y, − 3 x + 4y) (d) Write down the standard matrix representation of T. (e) Compute [T ]B. (f) Find [T (u)]B. (g) Calculate [T (b)]B for each of the basis vectors b ∈ B. Hence describe the geometric effect of the linear transformation T. Q2. Let the change of basis matrix from B = {b 1 , b 2 , b 3 } to C = {c 1 , c 2 , c 3 } be

PC,B =

(a) Find the coordinate vector of v = b 1 + 2b 2 + b 3 in basis C. (b) Find the coordinate vectors of the vectors b 1 , b 2 , b 3 in basis C. (c) Check that (^)  

− 1

Hence write down PB,C (d) Using PB,C , compute the coordinate vector of c 1 + 7c 3 in basis B. Check that your answer is consistent with (a).

Q3. Let A =

(a) By calculating Av, verify that the vector v =

 (^) is an eigenvector of A. What is the corresponding eigenvalue?

(b) Explain why u =

 (^) is not an eigenvector of A.

Mathematics and Statistics 1 University of Melbourne

MAST10007 Linear Algebra

Q4. Here is an image of the clock from the Old Arts clocktower on Parkville campus:

and here is the same image after applying a linear transformation T : R^2 → R^2 :

Identify the eigenvectors and their eigenvalues of the linear transformation T. Q5. Let T : R^2 → R^2 be the linear transformation given by orthogonal projection onto the line y = x. Without doing any calculations: (a) Give a non-zero vector v such that T (v) = v. (b) Give a non-zero vector v such that T (v) = 0. (c) List the eigenvalues of T and describe their eigenspaces.

Mathematics and Statistics 2 University of Melbourne