09-240/Classnotes for Tuesday September 29

From Drorbn
Jump to: navigation, search
WARNING: The notes below, written for students and by students, are provided "as is", with absolutely no warranty. They can not be assumed to be complete, correct, reliable or relevant. If you don't like them, don't read them. It is a bad idea to stop taking your own notes thinking that these notes can be a total replacement - there's nothing like one's own handwriting! Visit this pages' history tab to see who added what and when.

Vector subspaces

Definition. \mathbf W \subset \mathbf V is a "subspace" if it is a vector space under the operations it inherits from V.

Theorem. \mathbf W \subset \mathbf V is a subspace iff it is "closed under addition and vector multiplication by scalars", i.e. x, y \in \mathbf W \Rightarrow x + y \in \mathbf W and a \in F, x \in \mathbf W \Rightarrow ax \in \mathbf W.

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 \cup 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. \{0\}
  2. Any VS (which is a subspace of itself)
  3. A line passing through the origin (if it does not pass through the origin, then it is not closed under scalar multiplication)
  4. A plane
  5. Let \mathbf V = \mathbb M_{n \times n}(F). If W = \{ A \in \mathbf V : A^\top = A \}, then W is a subspace of V. (W is the set of "symmetric" matrices in V; AT denotes the transpose of A.)
  6. \mathbf W = \{ A \in \mathbb M_{n \times n} : \operatorname{tr}(A) = 0 \}
    where \operatorname{tr}(A) = \sum_{i=1}^n a_{ii} is the "trace" of A.
    Properties of trace:
    1. \operatorname{tr}(0 \cdot A) = 0
    2. \operatorname{tr}(cA) = c \cdot \operatorname{tr}(A)
    3. \operatorname{tr}(A + B) = \operatorname{tr}(A) + \operatorname{tr}(B)
    so W is indeed a subspace.

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

  1. W_1 \cap W_2 = \{ x \in \mathbf V : x \in \mathbf W_1 \mbox{ and } \mathbf W_2 \} is a subspace of V, W1, and W2.
  2. But W_1 \cup W_2 = \{ x \in \mathbf V : x \in \mathbf W_1 \mbox{ or } x \in \mathbf W_2 \} is a subspace of V iff \mathbf W_1 \subset \mathbf W_2 or \mathbf W_2 \subset \mathbf W_1. (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

u = a_1 u_1 + a_2 u_2 + \ldots + a_n u_n

Example: \mathbb P_n(F) = \{ \mbox{Polynomials of degree at most } n \mbox{ with coefficients in } F \}
= \left\{ \sum_{i=0}^n a_i x^i : a_i \in F \right\}

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

Example: Let \mathbf V = \mathbb M_{n \times n}(\real)
Let M_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, M_2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, M_3 = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, M_4 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
Then S = \{ M_1, M_2, M_3, M_4 \} generates V.

Proof: Given \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb M_{2 \times 2}(\real), write
\begin{pmatrix} a & b \\ c & d \end{pmatrix} = aM_1 + bM_2 + cM_3 + dM_4.

Example: Let N_1 = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, N_2 = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, N_3 = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, N_4 = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}
Does \{ N_1, N_2, N_3, N_4 \} generate V?

M_1 = -\frac23 N_1 + \frac13(N_2 + N_3 + N_4)
M_2 = -\frac23 N_2 + \frac13(N_1 + N_3 + N_4)
M_3 = -\frac23 N_3 + \frac13(N_1 + N_2 + N_4)
M_4 = -\frac23 N_4 + \frac13(N_1 + N_2 + N_3)

Then \begin{pmatrix} a & b \\ c & d \end{pmatrix} = aM_1 + bM_2 + cM_3 + dM_4 =

a \cdot \left( -\frac23 N_1 + \frac13(N_2 + N_3 + N_4) \right) + b \cdot \left( -\frac23 N_2 + \frac13(N_1 + N_3 + N_4) \right) + \ldots

Theorem: If S \in \mathbf V, then \operatorname{span}(S) = {all l.c. of elements of S} is a subspace of V.