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, where is notation for the norm or length of .
Distance between two points/vectors:
If and is the angle between them, then
Law of Cosines
Triangle defined by , , and
The above can be simplified to the form
is a unit vector iff
Unit vector in direction of is given by
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 ,
Two subspaces and in are said to be orthogonal if for all and .
is subspace, , iff for .
This means that is perpendicular to the th row of
Orthogonal complement given by
Let be the -axis.
is the -plane
- If , then
- If is a subspace of , then is also a subspace of
- Proof by contradiction. — CONTRADICTION!
- , so . , so subspace defined by
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
Fundamental Subspace Theorem