MATH 323 Lecture 19

From Notes
Jump to navigation Jump to search

« previous | Thursday, November 1, 2012 | next »


, where is notation for the norm or length of .

Distance between two points/vectors:

Theorem 5.1.1

If and is the angle between them, then

Law of Cosines

Triangle defined by , , and

The above can be simplified to the form

Unit Vector

is a unit vector iff

Unit vector in direction of is given by


Cauchy-Schwartz Inequality

Equality iff one of the vectors is zero or one of the vectors is a multiple of another ()


and are orthogonal (written if

Scalar and Vector Projections

Given two noncolinear vectors and

Let be a unit vector in the direction of and be the vector projection of in the direction of , where is the scalar projection of in the direction of


Given a point and a line , find the point on the line closest to .

Find vector on the line:

Take vector projection of onto :

Planes in 3D Space

(See MATH 251 Lecture 5#Planes→)

Generalization of Pythagorean Theorem

Given two orthogonal vectors and in ,

Orthogonal Subspaces

Two subspaces and in are said to be orthogonal if for all and .

is matrix

is subspace, , iff for .

This means that is perpendicular to the th row of


Orthogonal Complement

Orthogonal complement given by


Let be the -axis.

is the -plane


  1. If , then
  2. If is a subspace of , then is also a subspace of


  1. Proof by contradiction. — CONTRADICTION!
  2. , so . , so subspace defined by

Fundamental Subspaces


for some iff is in col space of .

We can write as a linear transformation .

The column space of is also called the range of :

Range of transpose matrix

Thus , and , and

Theorem 5.2.1

Fundamental Subspace Theorem