09-240/Classnotes for Tuesday September 29

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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; A^T 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, ..., u_n if there exists scalars a₁, ..., a_n such that

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

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

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.