Math 20F
Test 2
100 points
February 25, 2009
--------------------------------------------------------------------------------------------------Notation: All scalars and matrix entries and polynomial coefficents are in the set R of
real numbers.
Points: Problems (1), (3), (5), (6ii) 18 pts each ; (2), (4) 10 pts each ; (6i) 8 pts
Directions: Show all work and justify all answers.
----------------------------------------------------------------------------------------------------(1) Find a basis for the column space CS(A), where A denotes the 4 by 4 matrix
1 2 4 6
2 4 3 2
3 6 4 2
4 8 8 8
------------------------------------------------------------------------------------------------------(2) True or false: The determinant of the 4 by 4 matrix A in problem (1) equals zero.
------------------------------------------------------------------------------------------------------(3) Let A be a particular 3 by 3 matrix whose column space CS(A), viewed
geometrically, is a line through the origin. True or False: The null space NS(A) of this
matrix A, viewed geometrically, must be a plane through the origin.
-------------------------------------------------------------------------------------------------------(4) Let V be the vector space over R consisting of all polynomials in x whose degree
is at most 7 (including all constant polynomials such as 0, 1, etc.). For example, the
polynomial x5 - 2x3 + x + 4 is in V because it has degree 5, but the polynomial
x8 - 2x3 + x + 4 is not in V, because it has degree 8.
Find the dimension of the vector space V.
---------------------------------------------------------------------------------------------------------(5) Consider the 3 by 2 matrix A and the 3 by 1 vector b defined by
2 1
b1
A = -4 3 ,
b = b2
.
-6 4
b3
Show that b is in the column space CS(A) if and only if b3 = (7 b2 - b1 ) / 5
.
-----------------------------------------------------------------------------------------------------------(6) Suppose that the 2 by 2 matrix
0 1
2 0
represents a linear transformation T from R2 to R2
with respect to a certain basis v1 , v2 for R2 .
(i) True or false: T(v2) = v1 .
(ii) True or false: The 2 by 2 matrix
0 2
1 0
represents the transformation T above with respect to the new basis
v1 + v2 , v1 + 2v2 .