Terms  Definitions 

v 
length of v, defined by (v.v)^.5

dimNulA 
number of free variables of A

geometric multiplicity 
the dimension of the eigenspace corresponding to L, ie, dimNul(A  LI)

diagonalizable 
a matrix is diagonalizable if A is similar to a diagonal matrix, ie if A = PDP1 where P is invertible and D is a diagonal matrix

Chapter 5 Theorem 2: Eigenvectors corresponding to distinct eigenvalues... 
are linearly independent

subspace 
a subset of a vector space that:a. has the zero vectorb. is closed under vector additionc. is closed under scalar multiplication

rank 
the dimension of the column space of A

vector space 
a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars, subject to ten axioms.

null space 
the set of all solutions to the homogeneous equation Ax = 0

Algebraic multiplicity 
the number of times Li occurs as a root of the characteristic equation

similar 
A is similar to B if there is an invertible matrix P such that P(1)AP = B or, equivalently, A = PBP(1)

Process for QR factorization 
A = QRfind orthogonal basis for ColA = Span{x1, x2, x3}normalize vectorsQtA = R

Chapter 6 Theorem 7: properties of orthonormal matrices 
Ux = x(Ux).(Uy) = x.y(Ux).(Uy) = 0 iff x.y = 0

Orthogonal set 
A set of vectors {u1, u2, ... , up} in Rn is called an orthogonal set if ui.uj = 0 whenever i != j

Orthogonal projection of y onto u 
Y = (y.u)/(u.u) x uY is yhat

What is the characteristic equation and polynomial? 
equation: det(A  LI) = 0polynomial: det(A  LI)

The null space of an mxn matris is a subspace of... 
Rn.(the set of all solutions to a system Ax = 0 of m homogeneous linear equations in n unknowns is a subspace of Rn)

What are the properties of the direct sum? 
for an element u in U (U1....Um), the only way to write 0 is if all u in U = 0 and if V = u1+u2+...+um

What is a linear combination of the vectors (v1,...,vn) 
a1v1+a2v2+...+anvn

What is the span of (v1,...,vn) 
The set of all linear combinations. ie (7, 2, 9) = 2(2, 1, 3) + 3(1, 0, 1). hence (7,2,9) is in the span of ((2,1,3),(1,0,1))

When is a vector space finite dimensional 
If there exists a list of vectors which span the space.

What is linear independence 
It is linearly independent if there exists a1..an such that a1v1+a2v2+...+anvn = 0 is only true when all a = 0

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