Basis
A 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 Test
1. 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 Basis
P[x]_B' = [x]_B 
[x]_B' = P^{1} [x]_B 
[B B'] > [ I P^{1} ] 
[B' B ] > [ I P ] 
Cross Product
if 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 Space
u + 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 Matrices
A 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 
Eigenshit
The 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 Orthonormalization
1. 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 Spaces
R^{n} 
C(inf, +inf) 
C[a, b] 
P 
P_n 
M_m,n 
Inner Products
u = 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 
Kernal
For 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 Combo
v is a linear combo of u_1 ... u_n 
. 
Linear Independence
a 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 Transformation
V & 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) 
NonHomogeny
If xp is a solution to Ax = b then every solution to the system can be written as x = xp 


Nullity
Nullspace(A) = {x ε R^{n} : Ax = 0
Nullity(A) = dim(Nullspace(A))
= n  rank(A) 
Orthogonal Sets
Set 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 Onto
T 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 T
nullity(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 Matrix
Rank(A) = dim(Rspace) = dim(Cspace) 
Similar Matrices
For 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 Sets
S = {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 Subspace
1. u+v are in W 
2. cu is in w 

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