=========================preview======================
(MATH2121)[200X](f)quiz~=lqj1u^_20626.pdf
Back to MATH2121 Login to download
======================================================
MATH 2121 Linear Algebra Quiz 2 for T2a
Name:
Student ID: Time allowed: 20 minutes

1.
Determine whether the following matrices are diagonalizable. Explain your claims brie.y.


2.
Consider the function space V with a common domain D = [0,



3 0 4 5
A = 2 1 , B = 0 4 .
[Computational details are not required.]


4
]. Is the function f(x) = tan x

a linear combination of functions in S = {sin x, cos x}? Why?
END

1.
A is 2 2, and has 2 distinct eigenvalues =1, 3. So A is diagonalizable.

B has 1 eigenvalue = 4 and dim Nul (A . 4I) = 1, so B is not diagonalizable as there are not enough linearly independent eigenvectors of B (we need 2).

2.
No. We try to solve c1,c2 from the functional equation:


tan x = c1 sin x + c2 cos x, for all x [0,

In particular, we should have equalities when x = 0, x =


6
, and x =

4
:

. .. ..

3
11
1= c1 + c2
22
But the above system is inconsistent. So no such c1,c2 exist and hence tan x is not a linear combination of {sin x, cos x}.
MATH 2121 Linear Algebra Quiz 2 for T2b
Name:
Student ID: Time allowed: 20 minutes

1. Determine whether the following matrices are diagonalizable. Explain your claims brie.y.

. 1 1 . . 2 .1 .
A = 1 1 , B = 0 2 .

[Computational details are not required.]
x
2. Consider the function space V with a common domain D = R. Is the function f(x)= ea linear combination of functions in S = {sin x, cos x}? Why?
END
1.
A is 2 2, and has 2 distinct eigenvalues =0, 2. So A is diagonalizable.

B has 1 eigenvalue = 2 and dim Nul (A . 2I) = 1, so B is not diagonalizable as there are not enough linearly independent eigenvectors of B (we need 2).

2.
No. We try to solve c1,c2 from the functional equation:


x
e = c1 sin x + c2 cos x, for all x R.

In particular, we should have equalities when x = 0, x = , and x = :
2
.
. 1= c1 0+ c2 1 e 2 = c1 1+ c2 0

. e= c1 0+ c2 (.1)
x
But the above system is inconsistent. So no such c1,c2 exist and hence eis not a linear combination of {sin x, cos x}.