# Day 21 Outline

Tuesday, April 17, 2018

# Coins

Ask a (legit) mathematical question in class, get a coin worth 2% on Final Exam (max 3).

# Exam 2 discussion

Weights 30% 30% 15% instead of 25% 25% 25%

Problem 6a: the matrix display the images of the basis vectors as its columns

# 6.1 Inner product, length, orthogonality, cont'd

inner product/dot product/othogonality

two views of Ax (column view, row view)

recall rows live in domain, columns live in codomain

## orthogonal subspaces

examples: 2 lines in R2, 2 lines in R3, line and plane in R3

how/where can orthogonal subspaces intersect? Could two planes be orthogonal subspaces?

orthogonal complements: how to compute (basis of)? Orthogonality as linear system of equations

mth309.nullbasis()

Example: HW problems

# 6.2 A (Discrete) orthogonal set

Def

Example: make one in R3 with two successive orthogonal complement calculations

Theorem 4: An orthogonal set of non-zero vectors is LI.

Proof: Suppose lin comb is 0, take dot product with each vector in turn to show all coefficients 0.

Def: orthogonal basis

Theorem 5: Coordinates wrt orthog basis are easy to compute: Uc = y, multiply from left by UT.

Def: orthonormal sets

Theorem 6: mxn matrix U has orthonormal columns means UTU = I.

Theorem 7: U has orthonormal columns (U "orthogonal"): Orthogonal transformation preserves: length, dot product (in particular orthogonality)