Chapter 2.3

Class 7

Find an example
(a) Four distinct nonzero vectors that span $R^3$
(a) Four distinct nonzero vectors that span $R^4$
(a) Four distinct nonzero vectors that do not span $R^3$
(a) Four distinct nonzero vectors that do not span $R^4$

Sample solution:

(b) $\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 &0 & 0 \\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1 \end{bmatrix}$

(d) $\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 0 &0 & 0 \\ 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \end{bmatrix}$

• Def: Let {${u_1,u_2,…,u_m}$} be a set of vectors in $R^n$. If the only solution to the vector equation $x_1u_1+x_2u_2+...+x_mu_m=0$ is the trivial solution $x_1=x_2=…=x_m=0$, then we say {${u_1,u_2,…,u_m}$} is linear independent.

• Otherwise, if there are constants $x_1,x_2,…,x_m$ (not all zero) such that $x_1u_1+x_2u_2+...+x_mu_m=0$ then {${u_1,u_2,…,u_m}$} is called linearly dependent.

• If $A=[u_1 u_2 … u_m]$ ~ B and B is in echelon form
• (a) {$u_1,u_2,…,u_m$} is linear independent when there is a pivot in every column (no free variables)
• (b) {$u_1,u_2,…,u_m$} is linear dependent when there is a free variable.

Class 8

The Unifying Theorem (Version 1)

• Let S = {$a_1,a_2,…,a_n$} be a set of n vectors in $R^n$ and A = [$a_1 a_2 … a_n$]. The following are equivalent
1. S span $R^n$
2. S is linearly independent
3. Ax=b has a unique solution for every b in $R^n$

1 and 2, 2 and 3 are equivalent but not 1 and 2