BasisA set S is a basis for V if  1. S spans V  2. S is LI. 
If S is a basis for V then every vector in V can be written in one and only one way as a linear combo of vectors in S and every set containing more than n vectors is LD. Basis Test1. If S is a LI set of vectors in V, then S is a basis for V  2. If S spans V, then S is a basis for V 
Change of BasisP[x]_B' = [x]_B  [x]_B' = P^{1} [x]_B  [B B'] > [ I P^{1} ]  [B' B ] > [ I P ] 
Cross Productif u = u1i + u2j + u3k  AND  v = v1i + v2j + v3k  THEN  u x v = (u2v3  u3v2)i  (u1v3 u3v1)j + (u1v2  u2v1)k 
Definition of a Vector Spaceu + v is within V  u+v = v+u  u+(v+w) = (u+v)+w  u+0 = u  uu = 0  cu is within V  c(u+v) = cu+cv  (c+d)u = cu+du  c(du) = (cd)u  1*u = u 
  Diagonalizable MatricesA is diagonalizable when A is similar to a diagonal matrix.
That is, A is diagonalizable when there exists an invertible matrix P such that P^{1}AP is a diagonal matrix 
Dot Products Etc.length/norm v = sqrt(v_1^{2} +...+ v_n^{2}  cv = c v  v / v is the unit vector  distance d(u,v) = uv  Dot product u•v = (u_1v_1 +...+ u_nv_n)  n cos(theta) = u•v / (u v)  u&v are orthagonal when dot(u,v) = 0 
EigenshitThe scalar lambda(Y) is called an Eigenvalue of A when there is a nonzero vector x such that Ax = Yx.  Vector x is an Eigenvector of A corresponding to Y.  The set of all eigenvectors with the zero vector is a subspace of R^{n} called the Eigenspace of Y.  1. Find Eigenvalues: det(YI  A) = 0  2. Find Eigenvectors: (YI  A)x = 0  If A is a triangular matrix then its eigenvalues are on its main diagonal 
GramSchmidt Orthonormalization1. B = {v1, v2, ..., vn}  2. B' = {w1, w2, ..., wn}:   w1 = v1  w2 = v2  projw1v2  w3 = v3  projw1v3  projw2v3  wn = vn  ...   3. B'' = {u1, u2, ..., un}:   ui = wi/wi   B'' is an orthonormal basis for V  span(B) = span(B'') 
  Important Vector SpacesR^{n}  C(inf, +inf)  C[a, b]  P  P_n  M_m,n 
Inner Productsu = sqrt<u,u>  d(u,v) = uv  cos(theta) = <u,v> / (u v)  u&v are orthagonal when <u,v> = 0  proj_v u = <u,v>/<v,v> * v 
KernalFor T:V>W The set of all vectors v in V that satisfies T(v)=0 is the kernal of T. ker(T) is a subspace of v.
For T:R^{n} >R^{m} by T(x)=Ax ker(T) = solution space of Ax=0 & Cspace(A) = range(T) 
Linear Combov is a linear combo of u_1 ... u_n  . 
Linear Independencea set of vectors S is LI if c1v1 +...+ ckvk = 0 has only the trivial solution.
If there are other solutions S is LD. A set S is LI iff one of its vectors can
be written as a combo of other S vectors. 
Linear TransformationV & W are Vspaces. T:V>W is a linear transformation of V into W if:  1. T(u+v) = T(u) = T(v)  2. T(cu) = cT(u) 
NonHomogenyIf xp is a solution to Ax = b then every solution to the system can be written as x = xp 
  NullityNullspace(A) = {x ε R^{n} : Ax = 0
Nullity(A) = dim(Nullspace(A))
= n  rank(A) 
Orthogonal SetsSet S in V is orthogonal when every pair of vectors in S is orthogonal. If each vector is a unit vector, then S is orthonormal 
OnetoOne and OntoT is onetoone iff ker(T) = {0}  T is onto iff rank(T) = dim(W)  If dim(T) = dim(W) then T is onetoone iff it is onto 
Rank and Nullity of Tnullity(T) = dim(kernal)  rank(T) = dim(range)  range(T) + nullity(T) = n (in m_x n)  dim(domain) = dim(range) + dim(kernal) 
Rank of a MatrixRank(A) = dim(Rspace) = dim(Cspace) 
Similar MatricesFor square matrices A and A' of order n, A' is similar to A when there exits an invertible matrix P such that A' = P^{1} AP 
Spanning SetsS = {v1...vk} is a subset of vector space V. S spans V if every vector in v can be written as a linear combo of vectors in S. 
Test for Subspace1. u+v are in W  2. cu is in w 

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