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Linear Algebra Review
By Tim K. Marks UCSD
Virginia de Sa^ http://cs.gmu.edu/~kosecka/cs682.html Cogsci UCSD 108F Linear Algebra review
Vectors
The length of x, a.k.a. the norm or 2-norm of x, is e.g.,!
x = x 12 + x 22 + L + xn^2
!
x = 32 + 22 + 52 = 38
Good Review Materials
http://www.imageprocessingbook.com/DIP2E/dip2e_downloads/review_material_downloads.htm (Gonzales & Woods review materials)
Chapt Chapt. 1: Linear Algebra Review. 2: Probability, Random Variables, Random Vectors
Online vector addition demo: http://www.pa.uky.edu/~phy211/VecArith/index.html
Example (on board)
Inner product (dot product) of
two vectors
a =^ # $^ % % % "^623 & '^ ( ( ( !
b =^ " #^ $ $ $^415 % &^ ' ' '
!
a " b = aT^ b
!
= [ 6 2 " 3 ]^ # $^ % % %^415 & '^ ( ( (
Inner (dot) Product
vv αα^ uu The inner product is aThe inner product is a^ SCALAR.SCALAR.
Matrix Product
Product: Product:
Examples: Examples:
A and B must haveA and B must have compatible dimensionscompatible dimensions
Matrix Multiplication is not commutative:Matrix Multiplication is not commutative:
In Matlab: >> A*B
Matrix Sum
Sum:Sum:
Example:Example:
A and B must have theA and B must have the same dimensionssame dimensions
Determinant of a Matrix
Determinant: Determinant:
Example:Example:
A must be squareA must be square
Inverse of a 2D Matrix
Example:Example:
Inverses in Matlab
Other Terms
http://www.math.ubc.ca/~cass/courses/m309-8a/java/m309gfx/eigen.html
Eigenvalues^ Some Properties of and Eigenvectors
- If the corresponding eigenvectors λ 1 , …, λn are distinct eigenvalues e 1 , … of a matrix, then, en are linearly
- independent.A real, symmetric square matrix has real eigenvalues, with eigenvectors that can be chosen to be orthonormal.
Linear Independence
- A set of vectors is the vectors can be expressed as a linear linearly dependent if one of
combination of the other vectors. Example:
#^ $^ $^ $
&^ '^ '^ '^ ,
#^ $^ $^ $
&^ '^ '^ '^ ,
#^ $^ $^ $
- A set of vectors is linearly independent^ &^ '^ '^ ' if none
of the combination of the other vectors. vectors can be expressed as a linear
Example:
#^ $^ $^ $
&^ '^ '^ '^ ,
#^ $^ $^ $
&^ '^ '^ '^ ,
#^ $^ $^ $
&^ '^ '^ '
Linear Spaces
A linear space expressed as a linear combination of a set of basis is the set of all vectors that can be
vectors. We say this space is the vectors. span of the basis
- Example: spanned by each of the following two bases: R^3 , 3-dimensional Euclidean space, is
#^ $^ $^ $
&^ '^ '^ '^ ,
#^ $^ $^ $
&^ '^ '^ '^ ,
#^ $^ $^ $
&^ '^ '^ '
#^ $^ $^ $
&^ '^ '^ '^ ,
#^ $^ $^ $
&^ '^ '^ '^ ,
#^ $^ $^ $
&^ '^ '^ '
Linear Subspaces
A linear subspace of the vectors in a linear space. is the space spanned by a subset
- The space spanned by the following vectors is a two-dimensional subspace of R (^3).
#^ $^ $^ $
&^ '^ '^ '^ ,
#^ $^ $^ $
&^ '^ '^ '
#^ $^ $^ $
&^ '^ '^ '^ ,
#^ $^ $^ $
&^ '^ '^ '
What does it look like?
What does it look like?
- The space spanned by the following vectors is a two-dimensional subspace of R (^3).
Orthogonal and Bases Orthonormal
n span linearly independent real vectors Rn, n-dimensional Euclidean space
- They form a basis for the space.
- An orthogonal ai ⋅ aj = 0 basis, if a 1 i, …,j an satisfies
- An orthonormal ai ⋅ aj = 0 basis, if ai 1 , …j , an satisfies
- Examples.^ ai^ ⋅^ aj^ =^1 if^ i^ =^ j
Orthonormal Matrices
A square matrix is unitary) if its columns are orthonormal orthonormal (also called vectors.
- A matrix • If A is orthonormal A is orthonormal iff, A-1 (^) = AT AAT^ = I.
- A rotation^ AA Tmatrix is an^ =^ ATA^ =^ I. orthonormal matrix with determinant = 1. • It is also possible for an orthonormal matrix to have determinant = -1. This is a rotation plus a flip (reflection).
