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 sp
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
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
l
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
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 the
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 equat
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
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 fo
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 st
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
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 o