Study Guide and Worksheet for
Chapter 3
The basic concepts of this chapter are: linear combination, subspace,
span (noun and verb), linear dependence, basis, dimension, orthogonal
complement, row space, column space, null space, rank, vector space.
- Strive to understand the precise definitions of these terms and
their geometric connotations in R2 and R3;
compare with examples in text and class notes.
- Learn the computational techniques of the chapter, e.g., how to
use matrices to test for independence, etc, and be sure you understand
why they work.
- Learn how to use the theorems of the chapter to construct your
own examples, and to understand relationships among the basic concepts,
e.g., that every set of 4 vectors in R3 is dependent. (The
worksheet below should give you practice with this.)
- Review proof-type problems.
Fill in the following table. In each case, S refers to the set of all
linear combinations of the given vectors, and T refers to the
orthogonal complement of S.
Ambient
Space
|
Vectors
|
Independent?
|
Span ambient
space?
|
Basis for
ambient space?
|
Dim of S
|
Geom. Desc.
of S
|
Basis for T
|
R2
|
(1,2)
|
|
|
|
|
|
|
| R2 |
(1,2), (1,3)
|
|
|
|
|
|
|
| R2 |
(1,2), (1,3), (2,5)
|
|
|
|
|
|
|
| R2 |
(1,2), (2,4), (3,6)
|
|
|
|
|
|
|
| R2 |
(1,2), (2,4)
|
|
|
|
|
|
|
| R3 |
(0,0,0)
|
|
|
|
|
|
|
| R3 |
(1,1,1), (2,2,2)
|
|
|
|
|
|
|
| R3 |
(1,1,1), (1,2,1), (1,3,1)
|
|
|
|
|
|
|
| R3 |
(1,1,1), (1,1,2)
|
|
|
|
|
|
|
| R3 |
(1,1,1), (1,2,1), (2,4,2)
|
|
|
|
|
|
|
| R3 |
(1,1,1), (2,2,2), (4,4,4)
|
|
|
|
|
|
|
Fill in
the blanks.
a) Let A be an m x n matrix of rank r. The columns of
A are independent if _____. The cols of A span Rm if
_____. The cols of A are a basis for Rm if _____.
b) A linearly independent set in R4 cannot have
more than _____ vectors.
c) Two distinct vectors in R4 which cannot
together be part of a basis for R4 are __________.
True or False ?
a) The vector (2,3,0,1) can be expressed as a linear combination
of (1,0,0,0) and (0,1,0,1).
b) The solution set of 2x+y-z=0 is a subspace of R3.
c) The solution set of 2x+y-z=5 is a subspace of R3.
d) If v1, v2, v3 are dependent
vectors, then there are c1,c2,c3 in R
so that c1v1+c2v2+c3v3=0
and |c1|+|c2|+|c3|>0.
e) Any three vectors in R7 are independent.
f) A set which spans R3 can contain 5 vectors.
g) The set S:={(x,y,z) | x+y=z } is a subspace of R3.
h) The set S:={(x,y,z) | x2+y2=z2
} is a subspace of R3.
i) The set S:={(x,y,z) | x2+y2+z2
= 0 } is a subspace of R3.
j) There is a set of 5 vectors in R7 which is
independent.
k) Any set of 4 vectors in R4 which is independent
forms a basis for R4.
l) There is a 3-dimensional subspace of R5.