Matrix AlgebraA^{T} = A  (kA)^{T} = k(A)^{T}  (A+B)^{T} = A^{T} + B^{T}  (AB)^{T} = B^{T}A^{T}  tr(A^{T}) = tr(A)  tr(AB) = tr(BA)  u^{T}v = tr(uv^{T})  tr(uv^{T}) = tr(vu^{T})  (A^{T})ij = Aji  tr(A) = a11 + a22 ... + ann 
Elementary Matrixelementary matrices are invertible   A > RREF = I  A can be express as a product of E  A is invertible  Ax = 0 has only the trivial solution  Ax = b is consistent for every vector b in R^{n}  Ax = b has eactly 1 solution for every b in R^{n} If not, then all no.  A^{1} = Ek Ek1 ... E2 E1  [ A  I ] > [ I  A^{1} ] (how to find inverse of A)  Ax = b; x = A^{1}b 
ConsistencyEAx = Eb > Rx = b' , where b' = Eb  (Ax=b) [ A  b ] > [ EA  Eb ] (Rx = b') (but treat b as unknown: b1, b2...)  For it to be consistent, if R has zero rows at the bottom, b' that row must equal to zero 
Homogeneous SystemsLinear Combination of the vectors: v = c1v1 + c2v2 ... + cnvn (use matrix to find c)  Ax = 0  Homogeneous  Ax = b  Nonhomogenous  x = x0 + t1v1 + t2v2 ... + tkvk  Homogeneous  x = t1v1 + t2v2 ... + tkvk  Nonhomogeneous  xp is any solution of NH system and xh is a solution of H system  x = xp + xh 
Determinantsdet(A) = a1jC1j + a2jC2j ... + anjCnj  expansion along jth column  det(A) = ai1Ci1 + ai2Ci2 ... + ainCin  expansion along the ith row  Cij = (1)^{i+j} Mij  Mij = deleted ith row and jth column matrix   pick the one with most zeros to calculate easier  det(A^{T}) = det(A)  det(A^{1}) = 1/det(A)  det(AB) = det(A)det(B)  det(kA) = k^{n}det(A)   A is invertible iff det(A) not equal 0   det of triangular or diagonal matrix is the product of the diagonal entries  det(A) for 2x2 matrix  ad  bc 
  Examples of SubspacesIF: w1, w2 are within S  then w1+w2 are within S and kw1 is within S   the zero vector 0 it self is a subspace   R^{n} is a subspace of all vectors   Lines and planes through the origin are subspaces   The set of all vectors b such that Ax = b is consistent, is a subspace   If {v1, v2, ...vk} is any set of vectors in R^{n}, then the set W of all linear combinations of these vector is a subspace  W = {c1v1 + c2v2 + ... ckvk}; c are within real numbers 
Span the span of a set of vectors { v1, v2, ... vk} is the set of all linear combinations of these vectors  span { v1, v2, ... vk} = { v11t, t2v2, ... , tkvk}  If S = { v1, v2, ... vk}, then W = span(S) is a subspace  Ax = b is consistent if and only if b is a linear combination of col(A) 
Linear Independent if unique solution for a set of vectors, then it is linearly independent  c1v1 + c2v2 ... + cnvn = 0; all the c = 0   for dependent, not all the c = 0  Dependent if:  a linear combination of the other vectors  a scalar multiple of the other  a set of more than n vectors in R^{n}  Independent if:  the span of these two vectors form a plane   list the vectors as the columns of a matrix, row reduce it, if many solution, then it is dependent   after RREF, the columns with leading 1's are a maxmially linearly independent subset according to Pivot Theorem 
Diagonal, Triangular, Symmetric MatricesDiagonal Matrices  all zeros along the diagonal  Lower Triangular  zeros above diagonal  Upper Triangular  zeros below the diagonal  Symmetric if:  A^{T} = A  SkewSymmetric if:  A^{T} = A 
Adjoint and Cramer's Ruleadj(A) = C^{T}  C^{T} = matrix confactor of A  A^{1} = (1/det(A)) adj(A)  adj(A)A = det(A) I  x1 = det(A1) / det(A)  x2 = det(A2) / det(A)  xn = det(An) / det(A)  det(A) not equal 0  An is the matrix when the nth column is replaced by b 
  Hyperplane, Area/Volumea hyperplane in R^{n}  a1x1 + a2x2 ... + anxn = b   can also written as ax = b  to find a^{perp}  ax = 0, find the span  if A is 2x2 matrix:  det(A) is the area of parallelogram  if A is 3x3 matrix:  det(A) is the volume of parallelepiped   subtract points to get three vectors, then make it to a matrix to find the area/volume 
Cross Productu x v = (u2v3  u3v2, u3v1  u1v3, u1v2  u2v1)  u x v = v x u  k(u x v) = (ku) x v = u x (kv)  u x u = 0  parallel vectors has 0 for c.p.  u (u x v) = 0  v (u x v) = 0  u x v is perpendicular to span {u, v}  u x v = u v sin(theta), where theta is the angle between vectors 
Complex Numbercomplex number  a + ib  (a + ib) + (c + id) = (a + c) + i(b + d)  (a + ib)  (c + id) = (a  c) + i(b  d)  (a + ib) (c + id) = (ac + bd) + i(ad + bc)  (a + bx) (c + dx) = (ac + bdx^{2}) + x(ad + bc)  i^{2} = 1  z = a + ib  z bar = a  ib  the length(magnitude) of vector z  z = sqrt(z x z bar) = sqrt(a^{2} + b^{2})  z^{1} = 1/z = z bar / z^{2}  z1 / z2 = z1z2^{1}  z = z (cos(θ) + i (sin(θ))  polar form (r = z)  z1z2 = z1 z2 (cos(θ1 + θ2) + i (sin(θ1 + θ2))  z1/z2 = z1 / z2 (cos(θ1  θ2) + i (sin(θ1  θ2))  z^{n} = r^{n}(cos(n θ) + i sin(n θ))  r = z  e^{i theta} = cos(θ) + i sin(θ)  e^{i pi} = 1  e^{i pi} +1 = 0  z1z2 = r1r2 e^{i (θ1 + θ2)}  z^{n} = r^{n} e^{i nθ}  z1 /z2 = r1 / r2 e^{i (θ1  θ2)} 
Eigenvalues and EigenvectorsAx= λx  det(λI  A) = (1)^{n} det(A  λI)  pa(λ) = 3x3: det(A  λI); 2x2: det(λI  A)   solve for (λI  A)x = 0 for eigenvectors  Work Flow:  form matrix  compute pa(λ) = det(λI  A)  find roots of pa(λ) > eigenvalues of A  plug in roots then solve for the equation 

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