# Linear Algebra Flashcards

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) 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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