Math 2250-4
Wed Oct 9
4.1 - 4.3 Concepts related to linear combinations of vectors.
Exercise 1) Vocabulary review (these need to be memorized!)
A linear combination of the vectors v1 , v2 , ... vn is
The span of v1 , v2 , ... vn is
The vectors v1 , v2 , ... vn are linearly independent iff
The vectors v1 , v2 , ... vn are linearly dependent iff
The vectors v1 , v2 , ... vn are called a basis for the space they span iff
, Keep recalling that for vectors in =m all linear combination questions can be reduced to matrix questions
because any linear combination like the one on the left is actually just the matrix product on the right:
a
a
c
1
21
:
a
a
11
m1
a
Cc
2
22
:
a
a
12
m2
a
C...C c
n
2n
0
a
mn
c a
Cc a
C...
a
c a
Cc a
C ...
a
1 11
1n
=
1 21
2 12
2 22
=
:
c a
1 m1
11
21
a
a
:
Cc a
2 m2
C...
a
m1
12
22
:
a
m2
... a
... a
:
... a
1n
2n
:
mn
c
c
1
2
:
c
n
, Now finish the last exercise in Monday's notes, as well as Tuesday's notes about vector spaces and
subspaces. There are some more subspace exercises in today's notes.
Exercise 1) Can you use matrix theory to show that the only subspaces of =2 are
(0) The single vector 0, 0 T , or
(1) A line through the origin, i.e. span u for some non-zero vector u , or
(2) All of =2 .
Exercise 2) Can you use matrix theory to show that the only subspaces of =3 are
(0) The single vector 0, 0, 0 T , or
(1) A line through the origin, i.e. span u for some non-zero vector u , or
(2) A plane through the origin, i.e.span u, v where u, v are linearly independent, or
(3) All of =3 .
Exercise 3) The number of vectors in a basis for a subspace is called the dimension of the subspace. What
are the dimensions of the subspaces in Exercise 1 and Exercise 2?
In Tuesday's notes, Exercise 6, we showed that W = span v1 , v2 , ..., vn is closed under addition and
scalar multiplication, and so is a subspace. This is an explicit way of exhibiting a subspace, because you
are saying what all the elements of the subspace are. The other common way in which subspaces are
described is implicitly, as solution sets to systems of homogeneous equations:
Theorem: Let Am # n . Let W be the solution space to the homogenous matrix equation
A x = 0.
Then W is a subspace.
Can you show that W is closed under addition and scalar multiplication, in order to verify that it is a
subspace?