Math 24
Spring 2012
Problems from Monday April 9
First some denitions. If W1 and W2 are two subspaces of V , we dene
W1 + W2 = cfw_w1 + w2 | w1 W1 & w2 W2 .
In other words, W1 + W2 is the collection of all vectors you can get by adding
an element of W1 to
Math 24
Spring 2012
Friday, March 30
1. In the last class we said that the subspaces of R2 are
(a) The zero subspace, cfw_(0, 0).
(b) Any line through the origin.
(c) The entire space R2 .
What are the subspaces of R3 ?
(You do not have to prove your answ
Math 24
Spring 2012
Wednesday, April 4
Sample Solutions
1. Find a set of linearly independent vectors whose span is the plane in
R3 with equation
3x 2y + z = 0.
Sample Solution:
Since this subspace is a plane, we need two non-collinear vectors to
span it.
Math 24
Spring 2012
Questions (mostly) from the Textbook
1. TRUE OR FALSE ?
(a) The zero vector space has no basis. (F)
(b) Every vector space that is generated by a nite set has a basis.
(T)
(c) Every vector space has a nite basis. (F)
(d) A vector space
Math 24
Spring 2012
Wednesday, April 11
(1.) TRUE or FALSE? In these exercises, V and W are nite-dimensional vector spaces
over the same eld F , and T is a function from V to W .
a. If T is linear, then T preserves sums and scalar products. (T)
b. If T (x
Math 24
Spring 2012
Friday, April 13
1. Assume V and W are nite-dimensional vector spaces with ordered bases and
respectively, and T and U are linear transformations from V to W .
TRUE or FALSE?
(a) For any scalar a, aT + U is a linear transformation fro
Math 24
Spring 2012
Monday, April 16
(1.) TRUE or FALSE? In each part, V , W , and Z denote nite-dimensional vector
spaces with ordered bases , and respectively, T : V W and U : W Z denote
linear transformations, and A and B denote matrices.
(a.) [U T ] =
Math 24
Spring 2012
Thursday, April 19
(1.) TRUE or FALSE?
(a.) Suppose that = cfw_x1 , x2 , . . . , xn and = cfw_x1 , x2 , . . . , xn are ordered bases for
a vector space and Q is the change of coordinate matrix that changes -coordinates into
-coordin
Math 24
Spring, 2012
Wednesday, May 2, Sample Solutions
(1.) TRUE or FALSE?
(a.) If (A |b ) is obtained from (A|b) by a nite sequence of elementary column operations,
then the systems Ax = b and A x = b are equivalent. (F)
(b.) If (A |b ) is obtained from
Math 24
Spring 2012
Friday, May 4
(1.) TRUE or FALSE?
(a.) The function det : Mnn (F ) F is a linear transformation. (F)
(b.) The determinant of a n n matrix is a linear function of each row of the matrix
when the other rows are held xed. (T)
(c.) If A Mn