This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Answer: The most general solution of A = w with c = 12 is x2 = −2 + x3 2 .
x
x3
0
1
(c) The basis is formed by pivot rows of either A or its reduced form. (But not augmented
form, that is, we need to ignore the last column in our calculation.) Thus, the basis can be
given either as {(1, 1, 0), (0, 2, 4)} or as {(1, 0, −2), (0, 1, 2)}.
1 Ex. 3. Let A be an 7 × 11 matrix with the rank r = 4. What is the dimension of the
following four spaces. (a) Column space C (A) of A. (b) Row space C (AT ) of A. (c) Null
space N (A) of A. (d) Leftnull space N (AT ) of A.
( pts) Solution: Answers: (a) = r = 4. (b) = r = 4. (c) = n − r = 11 − 4 = 7.
(d) = m − r = 7 − 4 = 3. Name “leftnull” comes from the fact that y is in N (AT ) when it
is a solution of AT = 0, which is equivalent to (AT )T = 0T , that is, to T A = 0.
y
y
y Ex. 4. Let A be an m × n matrix of rank r. Describe precisely the possible number of
solutions of a system A = under the following assumptions:
xb
(a) r < m and...
View
Full
Document
 Spring '08
 STAFF
 Math, Linear Algebra, Algebra

Click to edit the document details