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Mathematics 123 Mathematics Mathematics
Typology: Exercises
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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 (^)
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