Vector Worksheets With Answers - Math 125, Exam 3 Page 5
ADVERTISEMENT
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
194. (2 pts each) Mark each statement True or False. Justify each answer.
(a) Let A be an m n matrix. The column space is the set of all vectors that can be written
as Ax for some x.
Solution: TRUE. The column space is the set of vectors b that can written b = Ax. That
is b = x
a
+ x
a
+ . . . + x a .
1
1
2
2
(b) A null space is a vector space.
Solution: TRUE. The null space is a vector space since it is the span of a set of basis vectors.
(c) If S = span u
, u
, . . . , u , then u
, u
, . . . , u
is a basis for S.
1
2
1
2
Solution: FALSE. u
, u
, . . . , u
is a basis for S if and only if all the vectors in the set
1
2
are linearly independent.
(d) The columns of an invertible n
n matrix form a basis for
.
Solution: TRUE. This is the theorem on page 75 in the notes. If A is an n
n matrix, the
columns of A span all of
if and only if A has an inverse.
(e) If the null space of a 7
6 matrix is 5 dimensional, then the row rank of the matrix
is one (1).
Solution: TRUE. This is an application of the rank + nullity = the number of columns
theorem. Recall that the column rank equals the row rank and, by definition, the nullity is
the rank of the null space. Therefore, rank + 5 = 6. Hence, the column rank (or dimension
of the column space) is 1.
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education