Projection and the Gram-Schmidt Process

The goal this class is to find orthonormal basis for a subspace.

Example:

What is an orthonormal basis for the span of $\{(2,0),(1,1)\}$ ?
How would we do it with 3 vectors?

Definition: Let $u,v\in \mathbb{R}^n$ with $v$ nonzero. Then the projection of u onto v is given by ParseError: KaTeX parse error: Undefined control sequence: \proj at position 1: \̲p̲r̲o̲j̲_v u=\frac{u\cd….

Theorem: Let $u,v\in \mathbb{R}^n$ and $c$ be a nonzero scalar. Then

ParseError: KaTeX parse error: Undefined control sequence: \proj at position 1: \̲p̲r̲o̲j̲_v u is in ParseError: KaTeX parse error: Undefined control sequence: \spn at position 1: \̲s̲p̲n̲{v}.
ParseError: KaTeX parse error: Undefined control sequence: \proj at position 3: u-\̲p̲r̲o̲j̲_v u is orthogonal to $v$
if $u$ is in ParseError: KaTeX parse error: Undefined control sequence: \spn at position 1: \̲s̲p̲n̲{v}, then ParseError: KaTeX parse error: Undefined control sequence: \proj at position 3: u=\̲p̲r̲o̲j̲_v u.
ParseError: KaTeX parse error: Undefined control sequence: \proj at position 1: \̲p̲r̲o̲j̲_v u = \proj_cv….

Let $S$ be a nontrivial subspace with orthogonal basis $\{v_1,\ldots,v_k\}$ . Then the projection of $u$ onto $S$ is given by ParseError: KaTeX parse error: Undefined control sequence: \proj at position 1: \̲p̲r̲o̲j̲_S u=\sum\sb{i=….

Theorem: Let $S$ be a nonzero subspace of $\mathbb{R}^n$ with orthogonal basis $\{v_1,\ldots,v_k\}$ , and let $u$ be a vector in $\mathbb{R}^n$ . Then

ParseError: KaTeX parse error: Undefined control sequence: \proj at position 1: \̲p̲r̲o̲j̲_S u is in $S$ .
ParseError: KaTeX parse error: Undefined control sequence: \proj at position 3: u-\̲p̲r̲o̲j̲_S u is orthogonal to $S$ .
if $u$ is in $S$ , then ParseError: KaTeX parse error: Undefined control sequence: \proj at position 3: u=\̲p̲r̲o̲j̲_S u.
ParseError: KaTeX parse error: Undefined control sequence: \proj at position 1: \̲p̲r̲o̲j̲_S u is independent of the choice of orthogonal basis for $S$ .

Theorem: (The Gram-Schmidt Process) Let $S$ be a subspace with basis $\{s_1,\ldots,s_n\}$ . Define $v_1,\ldots,v_k$ by

$v_1=s_1$ ,
ParseError: KaTeX parse error: Undefined control sequence: \proj at position 9: v_2=s_2-\̲p̲r̲o̲j̲\sb{v_1}s_2,
ParseError: KaTeX parse error: Undefined control sequence: \proj at position 9: v_3=s_3-\̲p̲r̲o̲j̲\sb{v_1}s_3-\pr…,
...
ParseError: KaTeX parse error: Undefined control sequence: \proj at position 9: v_k=s_k-\̲p̲r̲o̲j̲\sb{v_1}s_k-\ld…

Then $\{v_1,\ldots,v_k\}$ is an orthogonal basis. To make it orthonormal, just normalize each element.

Example: Find an orthonormal basis for the subspace $(1,0,1,1), (0,2,0,3), (-3, -1, 1, 5)$ .

Projection and the Gram-Schmidt Process

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