Homework 3
Due: Wednesday, September 17
   
0 
 0



1. Consider the set of vectors
,
1  .
1


0
1



α


(a) Describe the span of the vectors symbolically, e.g.,  −α  : α , β ∈ R ;


β
(b) Sketch the span of the vectors; and
(c) Describe the geometric view of the span in words (e.g., “a plane including the z-axis
and the point (1, 1, 0) ”), since sketches can be a bit hard to follow sometimes.
    
2 
 1



2
4  .
2. Repeat Problem 1 with the set of vectors
,


3
6
v1 , v~2 , . . . , ~
vr ) ,
3. Suppose ~u, v~1 , v~2 . . . v~r ∈ Rn , ~u ∈ span(~
v1 , v~2 , . . . , v~r ), and λ ∈ R. Show that λ~u ∈ span(~
as well.
2
3
1
2
3
as a linear combination of
? If so, write
,
∈ span
4. Is
1
8
3
1
8
1
and
. If not, why not?
3
5. Consider the linear system


w
 x 
1 4 0 3
1

=
.
0 0 1 −2  y 
5
z
(a) Find one solution to this system.
(b) Write the general solution of this system as a linear combination of your solution in (a)
the solutions to the homogeneous version of this equation.
3
(c) What changes if the vector of constants (right side of the equation) is changed to
.
1
(d) What changes if the vector of constants is set to the zero vector?
6. Consider the set of all vectors of size 2 (corresponding to the geometric object R2 ).
(a) Give a set of vectors that span R2 .
(b) Now give a larger set of vectors with the same span.
Professor Dan Bates
Colorado State University
M369 Linear Algebra
Fall 2008
(c) From the previous part, it should be clear that we can make the set of vectors arbitrarily
large. But how small can we make it? In other words, how few vectors must we provide
in order to span R2 ?
Professor Dan Bates
Colorado State University
M369 Linear Algebra
Fall 2008