Basic Equations
Network Flows 
1. the flow in an arc is only in one directions 
2. flow into a node = flow out of a node 
3. flow into the network = flow out of the network 
Balancing Chemical Equations 
1. add x's before each combo and both side 
2. carbo = x1 + 2(x3), set as system, solve 
Matrix 
augmented matrix 
variables and solution(rhs) 
coefficient matrix 
coefficients only, no rhs 
Vectors, Norm, Dot Product
maginitude (norm) of vector v is v; v ≥ 0 
if k>0, kv same direction as v 
magnitude = kv 
if k<0, kv opposite direction to v 
magnitude = k v 
vectors in R^{n} (n = dimension) 
v = (v1, v2, ..., vn) 
v = P1P2 = OP2  OP1 
displacement vector 
norm/magnitude of vector v 
sqrt( (v1)^{2}+(v2)^{2}...) 
v = 0 iff v =0 
kv = k v 
unit vector u in same direct as v 
u = (1/ v) v 
e1 = (1,0...) ... en = (0,...1) in R^{n} 
standard unit vector 
d(u,v) = sqrt((u1v1)^{2} + (u2v2)^{2} ... (unvn)^{2}) = uv 
d(u,v) = 0 iff u = v 
u·v = u1v1 + u2v2 ...+unvn u v cos(θ) 
dot product 
u and v are orthogonal if u·v = 0 (cos(θ) = 0) 
a set of vectors is an orthogonal set iff vi·vj = 0,if i≠j 
a set of vectors is an orthonormal set iff vi·vj = 0,if i≠j, and vi = 1 for all i 
(u·v)^{2} ≤ u^{2}v^{2} or u·v ≤ u v 
CauchySchwarz Inequality 
d(uv) ≤ d(u,w) + d(w,v) u+v ≤ u + v 
Triangle Inequality 
v1 + v2 ... + vk = v1 + v2 ... + vk 
Lines and Planes
a vector equation with parameter t 
x = x0 + tv, ∞ < t < +∞ 
solutin set for 3 dimension linear equation is a plane 
if x is a point on this plane (pointnormal equation) 
n·(xx0) = 0 
A(xx0)+B(yy0)+C(zz0) = 0 
x0 = (x0,y0,z0), n = (A, B, C) 
general/algebraic equation 
Ax+By+Cz = D 
two planes are parallel if n1 = kn2, orthogonal if n1·n2 = 0 
Matrix Algebra, Identity and Inverse Matrix
(A + B)ij = (A)ij + (B)ij 
(A  B)ij = (A)ij  (B)ij 
(cA)ij = c(A)ij 
(A^{T})ij = (A)ji 
(AB)ij = ai1b1j + ai2b2j + ... aikbkj 
Inner Product (number) is u^{T}v = u·v, u and v same size 
Outer Product (matrix) is uv^{T}, u and v can be any size 
(A^{T})^{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}) 
tr(A) = a11 + a22 ... + ann 
(A^{T})ij = Aji 
Identity matrix is square matrix with 1 along diagonals 
If A is m x n, AꞮn = A and ꞮmA = A 
a square matrix is invertible(nonsingular) if: 
AB = Ɪ = BA 
B is the inverse of A 
B = A^{1} 
if A has no inverse, A is not invertible (singular) 
det(A) = ad  bc ≠ 0 is invertible 
if A is invertible: 
(AB)^{1} = B^{1}A^{1} 
(A^{n})^{1} = A^{n} = (A^{1})^{n} 
(A^{T})^{1} = (A^{1})^{T} 
(kA)^{1} 
1/k(A^{1}), k≠0 
Elementary Matrix and Unifying Theorem
elementary matrices are invertible 
A^{1} = Ek Ek1 ... E2 E1 
[ A  Ɪ ] > [ Ɪ  A^{1} ] (how to find inverse of A) 
Ax = b; x = A^{1}b 
 A > RREF = Ɪ  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}  colum and rowvectors of A are linealy independent  det(A) ≠ 0  λ = 0 is not an eigenvalue of A  TA is one to one and onto If not, then all no. 
Consistency
EAx = 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 Systems
Linear 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 


Examples of Subspaces
IF: 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 Matrices
Diagonal 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 
Determinants
det(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 
Adjoint and Cramer's Rule
adj(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/Volume
a 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 Product
u 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 Number
complex 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 Eigenvectors
Ax= 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 
Linear Transformation
f: R^{n} > R^{m}, n = domain, m = codomain f(x1, x2, ...xn) = (y1, ...ym) 
T: R^{n} > R^{m} is a linear transformatin if 1. T(cu) = cT(u) 2. T(u +v) = T(u）+ T(v) 
for any linear transformation, T(0) = 0 
Rθ = [T(e1) T(e2)] = [cosθ −sinθ] [sinθ cosθ] 
matrix for rotation 
reflection across yaxis: T(x, y) = (x, y) 
reflection across xaxis: T(x, y) = (y, x) 
reflection across diagonal y = x, T(x, y) = (y, x) 
orthogonal projection onto the xaxis: T(x, y) = (x, 0) 
orthogonal projection onto the yaxis: T(x, y) = (0, y) 
u = (1/ v)v; express it vertically as u1 and u2 
A = [(u1)^{2} u2u1] [u1u2 (u2)^{2}] 
projection matrix 
contraction with 0 ≤ k < 1 (shrink), k > 1 (stretch) [x, y] > [kx, ky] 
compression in xdirection [x, y] > [kx, y] 
compression in ydirection [x, y] > [x, ky] 
shear in xdirection T(x,y) = (x+ky, y); [x+ky (1, k), y( 0, 1)] 
shear in ydirection T(x,y) = (x, y+kx); [x (1, 0), y (k, 1)] 
orthogonal projection on the xyplane: [x, y , 0] 
orthogonal projection on the xzplane: [x, 0 , z] 
orthogonal projection on the yzplane: [0, y , z] 
reflection about the xyplane: [x, y, z] 
reflection about the xzplane: [x, y, z] 
reflection about the yzplane: [x, y, z] 
Orthogonal Transformation
an orthogonal transformation is a linear transformation T; R^{n} > R^{n} that preserves lengths; T(u) = u 
T(u) = u <=> T(x)·T(y) = x·y for all x,y in R^{n} 
orthogonal matrix is square matrix A such that A^{T} = A^{1} 
1. if A is orthogonal, then so is A^{T} and A^{1} 
2. a product of orthonal matrices is orthogonal 
3. if A is orthogonal, then det(A) = 1 or 1 
4. if A is orthogonal, then rows and columns of A are each orthonormal sets of vectors 
Kernel, Range, Composition
ker(T) is the set of all vectors x such that T(x) = 0, RREF matrix, find the vector, ker(T) = span{(v)} 
the solution space of Ax = 0 is the null space; null(A) = ker(A) 
range of T, ran(T) is the set of vectors y such that y = T(x) for some x 
ran(T) = col([T]) = span{ [col1], [col2] ...}; Ax = b 
Important Facts: 1. T is one to one iff ker(T) = {0} 2. Ax = b, if consistent, has a unique solution iff null(A) = {0}; Ax = 0 has only the trivial solution iff null(A) = {0} 
Important facts 2: 1.T:R^{n} > R^{m} is onto iff the system Tx = y has a solution x in R^{n} for every y in R^{m} 2. Ax = b is consistent for every b in R^{m}(A is onto) iff col(A) = R^{m} 
The composition of T2 with T1 is: T2 ◦ T1 
(T2 ◦ T1)(x) = T2(T1(x)); T2 ◦ T1: R^{n} > R^{m} 
compostion of linear transformations corresponds to matrix application: [T2 ◦ T1] = [T1][T2] 
[T(θ1+θ2)] = [Tθ2] ◦ [Tθ1]; rotate then shear ≠ shear then rotate 
linear trans T: R^{n}>R^{m} has an inverse iff T is one to one, T^{1}: R^{m} > R^{n}, Tx = y <=> x = T^{1}y 
for Rn to Rn, [T^{1}] = [T]^{1}; [T]^{1}◦T = 1n <=> [T^{1}][T]=Ɪn 1n is identity transformation; Ɪn is identity matrix 
Basis, Dimension, Rank
S is a basis for the subspace V of R^{n} if: S is linearly idenpendent and span(S) = V 
dim(V) = k, k is the # of vectors 
row(A) = rows with leading ones after RREF 
col(A) = columns with leading ones from original A 
null(A) = free variable's vectors 
null(A^{T}) = after transform, the free variable vector 
The Rank Theorem: rank(A) = rank(A^{T}) for any matrix have the same dimension 
rank(A) = # of free vectors in span 
dim(row(A)) = dim(col(A)) = rank(A) 
dim(null(A)) = nullity(A) 
Orthogonal Compliment, DImention Theorem
S^{⟂} = {v ∈ R^{n}  v · w = 0 for all w ∈ S} 
S^{⟂} is a subspace of R^{n}; S^{⟂} = span(S)^{⟂} = W^{⟂} 
row(A)^{⟂} = null(A) 
null(A)^{⟂} = row(A) ((S^{⟂})^{⟂} = S iff S is subspace 
col(A)^{⟂} = null(A^{T}) 
null(A^{T})^{⟂} = col(A) 
The Dimension Theorem A is m x n matrix 
rank(A) + nullity(A) = n (k + (nk) = n) 
if W is a subspace of R^{n} 
dim(W) + dim(W^{⟂}) = n 

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