Math1020 Sgta Week6 Qns

MATH1020 ⋅ SESSION 1, 2024
SGTA ⋅ WEEK 6
SGTA questions for you

You should aim to be able to do these questions by the end of the SGTA, getting help where needed from fellow
students or instructors.
3 4 2
1. The 𝜆 = 2 eigenspace for the matrix [1 6 2] is two-dimensional. Find a basis for this eigenspace.
1 4 4
1 0
2. Find a matrix 𝑃 that diagonalises 𝐴 = [ ], and determine 𝑃 −1 𝐴𝑃.
6 −1
2 1 3 4
⎡ ⎤
⎢0 2 1 3⎥
3. Given that 𝜆 = 1 is an eigenvalue of the matrix ⎢ , find the corresponding eigenspace and a
⎢2 1 6 5⎥⎥
⎣1 2 4 8⎦
basis for the eigenspace.
4. For the following matrices, find a matrix 𝑃 that diagonalises 𝐴, and determine 𝑃−1 𝐴𝑃.
1 0 0 2 0 −2
−14 12
(𝑎) 𝐴 = [ ] (𝑏) 𝐴 = [0 1 1] (𝑐) 𝐴 = [0 3 0]
−20 17
0 1 1 0 0 3
0 0 1
5. Consider the invertible matrix 𝐵 = [ 2 0 0]. Determine 𝐵 2 and use the Cayley-Hamilton Theorem
−1 3 0
−1
to find 𝐵 .
6. Which of the following are linear transformations?
(a) 𝑇 ∶ ℝ2 → ℝ2 , where 𝑇(𝑥1 , 𝑥2 ) = (|𝑥1 |, |𝑥2 |).
(b) 𝑇 ∶ ℝ2 → ℝ2 , where 𝑇(𝑥1 , 𝑥2 ) = (2𝑥1 , −3𝑥2 ).
(c) 𝑇 ∶ ℝ4 → ℝ3 , where 𝑇(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) = (𝑥2 − 𝑥3 , 𝑥1 + 1, 𝑥2 ).
Further practice
Here are some questions for further practice, possibly after the SGTA.
7. Which of the following are linear transformations?

(a) 𝑇∶ ℝ3 → ℝ2 , where 𝑇(𝑥1 , 𝑥2 , 𝑥3 ) = (𝑥1 + 𝑥2 , 𝑥1 − 𝑥3 ).
(b) 𝑇∶ ℝ2 → ℝ2 , where 𝑇(𝑥1 , 𝑥2 ) = (𝑥1 − 𝑥2 , 1).
(c) 𝑇∶ ℝ4 → ℝ, where 𝑇(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) = (𝑥1 − 2𝑥2 + 𝑥3 − 𝑥4 ).
(d) 𝑇∶ ℝ2 → ℝ3 , where 𝑇(𝑥1 , 𝑥2 ) = (𝑥12 , 0, 𝑥2 ).
8. Suppose that the characteristic equation of some matrix 𝐴 is given by
(i) 𝑝(𝜆) = 𝜆(𝜆 − 1)2 (𝜆 + 2)2 ;
(ii) 𝑝(𝜆) = (𝜆 + 1)3 (𝜆 + 2)(𝜆 + 4)5 .
In each case:
(a) Determine the size of 𝐴.
(b) Is 𝐴 invertible? If it is invertible state the eigenvalues of 𝐴−1 .
(c) How many eigenspaces does 𝐴 have?
9. For each of the following matrices determine all eigenspaces 𝐸 𝜆 , and check if the multiplicity of 𝜆 equals
the dimension of 𝐸 𝜆
1 0 0 2 1 1
(a) 𝐴 = [0 3 2] (b) 𝐴 = [1 2 1]
0 2 3 1 1 2
10. Answer the following true or false, giving reasons for your answer.
(a) 𝜆5 − 2𝜆2 − 1 is be the characteristic polynomial of some 3 × 3 matrix.
(b) The complex number 3 + 𝑖 can be an eigenvalue.
(c) The eigenspaces of an 𝑛 × 𝑛 matrix are subspaces of ℝ𝑛 .
5 2 0 1 0
11. The matrix 𝐴 = [−4 −1 0] has eigenvalues 1, 2, 3 with corresponding eigenvectors [−2], [0] and
0 0 2 0 1
1
[−1]. (You do not need to verify this.)
0
(a) Write down a diagonal matrix 𝐷 and a matrix 𝑃 such that 𝑃−1 𝐴𝑃 = 𝐷.
(b) Hence, calculate 𝐴9 .
0 −2 −3
12. Consider the matrix 𝐴 = [−1 1 −1]
2 2 5
(a) Determine 𝐴2 .
(b) Use the Cayley-Hamilton Theorem to find 𝐴−1 .
13. Let 𝐹 ∶ ℝ3 → ℝ3 be a linear transformation such that
𝐹(1, 0, 0) = (1, 1, 0), 𝐹(0, 1, 0) = (0, 0, −1), 𝐹(1, 2, 3) = (1, 1, 1).
Find (i) 𝐹(1, 2, 3) and (ii) 𝐹(𝑥1 , 𝑥2 , 𝑥3 ).
14. Find the kernel of the linear transformations
(a) 𝑇 ∶ ℝ3 → ℝ3 such that 𝑇(𝑥1 , 𝑥2 , 𝑥3 ) = (𝑥1 , 𝑥1 , 0)
(b) 𝑇 ∶ ℝ4 → ℝ2 such that 𝑇(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) = (𝑥1 + 𝑥2 , 𝑥3 + 𝑥4 ).
Matlab problems
These are problems that require you to use Matlab.
5 8 16
15. Consider the matrix 𝐴 = [ 4 1 8].
−4 −4 −11
(a) Determine the characteristic polynomial of 𝐴 by using the command charpoly(A) in Matlab.
This command produces the coefficients of the characteristic polynomial.
(b) If you provide a symbolic variable as the second input to charpoly, Matlab will return the char-
acteristic polynomial written in terms of that variable. The commands
>> syms lambda
>> charpoly(A,lambda)
will return the characteristic polynomial of 𝐴.
The commands
>> syms x
>> solve(x^2-2==0,x)
solve the quadratic equation 𝑥2 − 2 = 0.
Use solve and charpoly to find all solutions to the characteristic polynomial of 𝐴.
2
(c) Check your answers to (b), by using eig(A).
1 −3 3
16. Use the Matlab command [P,D]=eig(A) to find a matrix 𝑃 that diagonalises 𝐴 = [3 −5 3]. Check
6 −6 4
−1
your answer by computing 𝑃 𝐴𝑃.
Additional problems
These are problems that students who would like something a little more challenging can try at home after the
SGTA. Your SGTA instructor may discuss some of these problems in the SGTA if time permits.
5 2 0
17. Consider again the matrix 𝐴 = [−4 −1 0] from Question 11.
0 0 2
(a) Find a matrix 𝐵 such that 𝐵 2 = 𝐴.
(b) How many such matrices 𝐵 are there?
18. Find the kernel and the range of the linear transformation 𝑇 ∶ ℝ3 → ℝ3 , where
𝑇(𝑥1 , 𝑥2 , 𝑥3 ) = (𝑥1 − 𝑥2 + 𝑥3 , 2𝑥1 − 𝑥2 − 𝑥3 , −𝑥1 − 2𝑥2 + 𝑥3 ).

Math1020 Sgta Week6 Qns

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MATH1020 ⋅ SESSION 1, 2024

SGTA questions for you

7. Which of the following are linear transformations?

𝑇(𝑥1 , 𝑥2 , 𝑥3 ) = (𝑥1 − 𝑥2 + 𝑥3 , 2𝑥1 − 𝑥2 − 𝑥3 , −𝑥1 − 2𝑥2 + 𝑥3 ).

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