Take Home Exam 1
Sample Solutions
Problem A.
Let T : R4 R3 be the linear transformation dened by
x1 + x3 2x4
.
2x2 x3 + x4
T (x1 , x2 , x3 , x4 ) =
2x1 + 2x2 + x3 3x4
(1) Find a basis for the null space N(T ).
Solution The equation T (x1 , x2 , x3 , x4 )
Homework Notes Week 2
Math 24 Spring 2014
1.4:12* Remember to use theorems (and to correctly reference them). Proving
everything from scratch every time takes a lot of time and eort!
In this case, the inclusion W span(W) is an immediate consequence of The
Homework Notes Week 1
M ATH 24 S PRING 2014
1.2#9* The most common error for this problem was to use Theorem 1.1 in a
non-literal manner. Theorem 1.1 gives only one of the four possible cancellation
laws:
1. If x + z = y + z then x = y.
2. If z + x = z +
Homework Notes Week 3
Math 24 Spring 2014
2.1:5 To see that T : P2 (R) P3 (R) is linear, it suces to check that
T (af (x) + bg(x) = aT (f (x) + bT (g(x)
for any f (x), g(x) P2 (R) and any real numbers a, b. From familiar properties of
polynomials from alg
Homework Notes Week 4
Math 24 Spring 2014
2.4#4 Let A and B be n n invertible matrices. We want to show that AB is
invertible and that (AB)1 = B 1 A1 .
Recall that an n n matrix X is invertible if there is an n n matrix Y with
XY = Y X = I, the n n identi
Quiz 2
M ATH 24 S PRING 2014
Sample Solutions
Find a basis for the subspace W of R3 consisting of all vectors (x1 , x2 , x3 ) such that x1 + 3x2 2x3 = 0.
Justify your answer.
Solution. The given equation can be rewritten x1 = 2x3 3x2 . Given x2 = a and x3
Quiz 4
M ATH 24 S PRING 2014
Sample Solutions
Given
1
A = 4
0
1
3
5
1
1
4 = 4
3
0
0
1
5
0
1
0 0
1
0
1
1
0
1
0
3
Compute A1 .
Solution. The colums of A1 are v1 = A1 e1 , v2 = A1 e2 and v3 = A1 e1 . In other words, they are the solutions
to the equations
Quiz 3
M ATH 24 S PRING 2014
Sample Solutions
The space P2 (R) has the standard ordered basis = cfw_1, x, x2 and the space R2 has the standard ordered basis
= cfw_e1 , e2 .
Let T : P2 (R) R2 be the linear transformation dened by T (f (x) = (f (0), f (1)
Quiz 4
M ATH 24 S PRING 2014
Sample Solutions
Given
0
A = 1
2
1
3
0
1
4
2
1
3
2
1
2 .
2
Knowing that the rst, second and fourth columns of A form a basis for R3 , write down the reduced row echelon form
of A. Justify your answer. (Hint: See Theorem 3.16.)
Quiz 3
M ATH 24 S PRING 2014
Sample Solutions
The space P2 (R) has for standard ordered basis = cfw_1, x, x2 and another ordered basis
=
1 2
2x
1
1 x, 1 x2 , 2 x2 + 1 x .
2
2
The space R3 has the standard ordered basis = cfw_e1 , e2 , e3 .
Let T : P2 (R
Quiz 1
M ATH 24 S PRING 2014
Sample Solutions
A n n matrix A over the eld R of real numbers is skew-symmetric if At = A. (The transpose At of a matrix A
is dened on page 17.) By Exercise 28 of Section 1.3, we know that the set Wn of all n n skew-symmetric
Homework Notes Week 5
Math 24 Spring 2014
3.1#8* Theorem. If a matrix Q can be obtained from a a matrix P by an
elementary row operation, then P can be obtained from Q by an elementary row
operation of the same type.
Proof. There are three types of elemen