Cheatography

# Linear Algebra - MATH 232 Cheat Sheet by fionaw

This is for math 232 - linear algebra, midterm 2

### 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 = k||v|| if k<0, kv opposite direction to v magnitude = |k| ||v|| vectors in Rn (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 Rn standard unit vector d(u,v) = sqrt((­u1-v1)2 + (u2-v2)2 ... (un-vn)2) = ||u-v|| 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|| Cauchy­-Sc­hwarz 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(point­-normal equation) n·(x-x0) = 0 A(x-x0­)+B­(y-­y0)­+C(­z-z0) = 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 (AT)ij = (A)ji (AB)ij = ai1b1j + ai2b2j + ... aikbkj Inner Product (number) is uTv = u·v, u and v same size Outer Product (matrix) is uvT, u and v can be any size (AT)T = A (kA)T = k(A)T (A+B)T = AT + BT (AB)T = BTAT tr(AT) = tr(A) tr(AB) = tr(BA) uTv = tr(uvT) tr(uvT) = tr(vuT) tr(A) = a11 + a22 ... + ann (AT)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-1A-1 (An)-1 = A-n = (A-1)n (AT)-1 = (A-1)T (kA)-1 1/k(A-1), k≠0

### Elementary Matrix and Unifying Theorem

 elementary matrices are invertible A-1 = Ek Ek-1 ...  E2 E1 [ A | Ɪ ] -> [ Ɪ | A-1 ] (how to find inverse of A) Ax = b; x = A-1b - 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 Rn - Ax = b has eactly 1 solution for every b in Rn - 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 Non-ho­mog­enous x = x0 + t1v1 + t2v2 ... + tkvk Homoge­neous x = t1v1 + t2v2 ... + tkvk Non-ho­mog­eneous xp is any solution of NH systemand 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 - Rn 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 Rn, 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 Rn 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: AT = A Skew-S­ymm­etric if: AT = -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(AT) = det(A) det(A-1) = 1/det(A) det(AB) = det(A)­det(B) det(kA) = kndet(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

 adj(A) = CT CT = 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 Rn a1x1 + a2x2 ... + anxn = b - can also written as ax = b to find aperp 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 + bdx2) + x(ad + bc) i2 = -1 z = a + ib z bar = a - ib the length­(ma­gni­tude) of vector z |z| = sqrt(z x z bar) = sqrt(a2 + b2) 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)) zn = rn(cos(n θ) + i sin(n θ)) r = |z| ei theta = cos(θ) + i sin(θ) ei pi = -1 ei pi +1 = 0 z1z2 = r1r2 ei (θ1 + θ2) zn = rn ei nθ z1 /z2 = r1 / r2 ei (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: Rn -> Rm, n = domain, m = co-domain f(x1, x2, ...xn) = (y1, ...ym) T: Rn -> Rm 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 y-axis: T(x, y) = (-x, y) reflection across x-axis: T(x, y) = (y, -x) reflection across diagonal y = x, T(x, y) = (y, x) orthogonal projection onto the x-axis: T(x, y) = (x, 0) orthogonal projection onto the y-axis: 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 x-dire­ction [x, y] -> [kx, y] compre­ssion in y-dire­ction [x, y] -> [x, ky] shear in x-dire­ction T(x,y) = (x+ky, y); [x+ky (1, k), y( 0, 1)] shear in y-dire­ction T(x,y) = (x, y+kx); [x (1, 0), y (k, 1)] orthogonal projection on the xy-plane: [x, y , 0] orthogonal projection on the xz-plane: [x, 0 , z] orthogonal projection on the yz-plane: [0, y , z] reflection about the xy-plane: [x, y, -z] reflection about the xz-plane: [x, -y, z] reflection about the yz-plane: [-x, y, z]

### Orthogonal Transf­orm­ation

 an orthogonal transf­orm­ation is a linear transf­orm­ation T; Rn -> Rn that preserves lengths; ||T(u)|| = ||u|| ||T(u)|| = ||u|| <=> T(x)·T(y) = x·y for all x,y in Rn orthogonal matrix is square matrix A such that AT = A-1 1. if A is orthog­onal, then so is AT 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:Rn -> Rm is onto iff the system Tx = y has a solution x in Rn for every y in Rm 2. Ax = b is consistent for every b in Rm(A is onto) iff col(A) = Rm The compos­ition of T2 with T1 is: T2 ◦ T1 (T2 ◦ T1)(x) = T2(T1(x)); T2 ◦ T1: Rn -> Rm 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: Rn->Rm has an inverse iff T is one to one, T-1: Rm -> Rn, Tx = y <=> x = T-1y 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 Rn 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(AT) = after transform, the free variable vector The Rank Theorem: rank(A) = rank(AT) 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 ∈ Rn | v · w = 0 for all w ∈ S} S⟂ is a subspace of Rn; S⟂ = span(S)⟂ = W⟂ row(A)⟂ = null(A) null(A)⟂ = row(A) ((S⟂)⟂ = S iff S is subspace col(A)⟂ = null(AT) null(AT)⟂ = col(A) The Dimension Theorem A is m x n matrix rank(A) + nullity(A) = n (k + (n-k) = n) if W is a subspace of Rn dim(W) + dim(W⟂) = n