09-240/Classnotes for Tuesday September 29

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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):

  1. A unit circle is not closed under addition of scalar multiplication.
  2. The x-axis y-axis is closed under scalar multiplication, but not under addition.
  3. A single quadrant of the Cartesian plane is closed under addition, but not under scalar multiplication.

Examples of subspaces:

  1. Any VS (which is a subspace of itself)
  2. A line passing through the origin (if it does not pass through the origin, then it is not closed under scalar multiplication)
  3. A plane
  4. Let . If , then W is a subspace of V. (W is the set of "symmetric" matrices in V; AT denotes the transpose of A.)
  5. where is the "trace" of A.
    Properties of trace:
    so W is indeed a subspace.

Claim: If W1 and W2 are subspaces of V, then

  1. is a subspace of V, W1, and W2.
  2. But is a subspace of V iff or . (See HW2 pp. 20-21, #19.)

Linear combinations

Definition: A vector u is a "linear combination" (l.c.) of vectors u1, ..., un if there exists scalars a1, ..., an such that


Definition: A subset "generates" or "spans" V iff the set of linear combinations of elements of S is all of V.

Example: Let
Then generates V.

Proof: Given , write

Example: Let
Does generate V?


Theorem: If , then {all l.c. of elements of S} is a subspace of V.