##### Linear Algebra set_1
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#### Complete List of Terms and Definitions for Linear Algebra set_1

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 = PDP-1 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 non-empty 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 y-hat
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)