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:R2→R2be 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
−5−2 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
