Span, Linear Independence, and Basis
Problem 1:
Look at Vector([2,1,1]) and let
L=Span of (2,1,1) = {t (2,1,1) where t is real}. Notice L is a line
through the origin in R^{3} and we can graph the vector
in Maple (without the arrowhead) as:
with(LinearAlgebra): with(plots):
a:=spacecurve({[2*t,1*t,1*t,t=0..1]},color=red):
display(a);
Part 1:
Find a vector w_{1} so that
{Vector([2,1,1]), w_{1}}
is a basis for some plane P_{1}.
Part 2:
Find a vector w_{2}
not on the same line through the origin as w_{1}
so that {Vector([2,1,1]), w_{2}} is also a basis for the same
plane P_{1}
Part 3: In Maple, use spacecurve commands and display
as above to show that all three vectors lie in the same plane but
no 2 are on the same line
[use different colors like black, blue, green..., and one display
command like display(a,b,c);]
Part 4:
Describe all the vectors w for which
{Vector([2,1,1]), w} is a basis for the same plane
P_{1} [Hint  linear combinations are in the same geometric space
so think about what linear combinations
a Vector([2,1,1]) + b w_{1}
you can use that will give a basis
ie what a's and b's you can use].
Part 5:
Find a vector u
so that {Vector([2,1,1]), u}
is a basis for a different plane P_{2} through the origin.
Part 6:
Add u to your graph from Part 3 to show it lies outside the
plane.
Problem 2: Let W be the subspace of R^{4}
spanned by the vectors u_{1}:=(1,2,3,4),
u_{2}:=(4,2,1,5), and
u_{3}:=(3,5,1,7).
Determine if the vectors v:=(8,9,5,16) and
w:=(7,2,1,3) are in W by setting up the augmented matrices and solving
using
M:=Matrix([[
ReducedRowEchelonForm(M).
Next, determine whether
{u_{1},
u_{2},
u_{3}, v}
and
{u_{1},
u_{2},
u_{3}, w}
are linearly independent sets by setting up the homogeneous
equations and solving.
Does the set of all five vectors span R^{4}?
If you finish before we come back together to go over these,
write down your answers for your notes..
Definitions Review Recall that
linear combinations are in the same geometric space.
A collection of vectors is a
basis if it both spans and is linearly
independent. For a plane, 2 linearly independent vectors (ie not on the
same line) will span and 2 vectors that span are also linearly independent.
The vectors
v_{1}, v_{2}, ..., v_{
n}
span
S if each element of S can be written as a linear combination
of these vectors. In other words, every s in S can be written as

s =
c_{1}v_{1} + c_{2}
v_{2} + ... + c_{n}v_{n}
for some constants c_{i}.
The vectors v_{1}, v_{2}, ..., v_{n} are linearly independent, if and only if the following condition is satisfied:
Whenever c_{1}, c_{2}, ..., c_{n} are elements of R such that:
 c_{1}v_{1} + c_{2}v_{2} + ... + c_{n}v_{n} = 0
then c_{i} = 0 for i = 1, 2, ..., n.