LINEAR INDEPENDENCE and BASES

Linear Independence:

$\{ {\bf v}_1, {\bf v}_2, \dots, {\bf v}_p \} \in V$ are linearly
independent if $c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_p{\bf v}_p = {\bf0}$ has only the trivial solution.

Characterization:

$\{ {\bf v}_1, {\bf v}_2, \dots, {\bf v}_p \}$ are linearly
dependent if some ${\bf v}_i$ is a linear combination of the other vectors.

Basis:

$B = \{ {\bf b}_1, {\bf b}_2, \dots, {\bf b}_p \} \in V$ is a basis for a subspace H of V if B is linearly independent and Span(B) = H.

• Rⁿ: standard basis is $\{{\bf e}_1, {\bf e}_2, \dots, {\bf e}_n\}$.
• P_n: standard basis is $\{1, t, t^2, \dots, t^n\}$.
• Pivot columns for A form a basis for Col(A).
• Spanning set for solutions of $A{\bf x}\ = {\bf0}$ is a basis for Nul(A).

Spanning Set Theorem:

Suppose $S = \{ {\bf v}_1, {\bf v}_2, \dots, {\bf v}_p \} \in V$ and H = Span(S).

1.

Removal of some dependent ${\bf v}_k$ from S gives a smaller spanning set for H.

2.

If $H \neq \{{\bf0}\}$, some subset of S is a basis for H.

COORDINATE SYSTEMS

Assume a basis $B = \{{\bf b}_1, {\bf b}_2, \ldots, {\bf b}_n \}$ for vector space V.

Uniqness:

for any ${\bf x}\in V$, ${\bf x}= c_1{\bf b}_1 + c_2{\bf b}_2 + \cdots + c_n{\bf b}_n$ uniquely.

Coordinates:

if ${\bf x}= c_1{\bf b}_1 + c_2{\bf b}_2 + \cdots + c_n{\bf b}_n$, then the c_i's are the B coordinates of ${\bf x}$; notation: $[{\bf x}]_B = [c_1, \dots, c_n]^T$.

Change-of-Coordinates Matrix:

For vectors in Rⁿ, the change-of-coordinates matrix P_B is defined by ${\bf x}= P_B[{\bf x}]_B$; $P_B= [{\bf b}_1\ {\bf b}_2\ \ldots\ {\bf b}_n]$.