09240/Classnotes for Tuesday September 29

Yangjiay  Page 1
Vector subspaces
Definition. is a "subspace" if it is a vector space under the operations it inherits from V.
Theorem. is a subspace iff it is "closed under addition and vector multiplication by scalars", i.e. and .
Goal: Every VS has a "basis", so while we don't have to use coordinates, we always can.
Examples of what is not a subspace (without diagrams):
 A unit circle is not closed under addition of scalar multiplication.
 The xaxis yaxis is closed under scalar multiplication, but not under addition.
 A single quadrant of the Cartesian plane is closed under addition, but not under scalar multiplication.
Examples of subspaces:
 Any VS (which is a subspace of itself)
 A line passing through the origin (if it does not pass through the origin, then it is not closed under scalar multiplication)
 A plane
 Let . If , then W is a subspace of V. (W is the set of "symmetric" matrices in V; A^{T} denotes the transpose of A.)

 where is the "trace" of A.
 Properties of trace:
 so W is indeed a subspace.
Claim: If W_{1} and W_{2} are subspaces of V, then
 is a subspace of V, W_{1}, and W_{2}.
 But is a subspace of V iff or . (See HW2 pp. 2021, #19.)
Linear combinations
Definition: A vector u is a "linear combination" (l.c.) of vectors u_{1}, ..., u_{n} if there exists scalars a_{1}, ..., a_{n} such that
Example:
Definition: A subset "generates" or "spans" V iff the set of linear combinations of elements of S is all of V.
Example: Let
Let
Then generates V.
Proof: Given , write
Example: Let
Does generate V?
Then
Theorem: If , then {all l.c. of elements of S} is a subspace of V.