Harvard College. Math 21b: Linear Algebra and Differential Equations Formula and Theorem Review

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1 Harvard College Math 21b: Linear Algebra and Differential Equations Formula and Theorem Review Tommy MacWilliam, 13 May 5,

2 Contents Table of Contents 4 1 Linear Equations Standard Representation of a Vector Reduced Row Echelon Form Elementary Row Operations Rank of a Matrix Dot Product of Vectors The Product A x Algebraic Rules for A x Linear Transformations Linear Transformations Scaling Matrix Orthogonal Projection onto a Line Reflection Matrix Rotation Matrix Shear Matrix Matrix Multiplication Invertibility Finding the Inverse Properties of Invertible Matrices Subspaces of R n and Their Dimensions Image of a Function Span Kernel Subspaces of R n Linear Independence Dimension Rank-Nullity Theorem Coordinates Linearity of Coordinates Matrix of a Linear Transformation Similar Matrices Linear Spaces Linear Spaces Finding a Basis of a Linear Space V Isomorphisms

3 5 Orthogonality and Least Squares Orthongality and Length Orthonormal Vectors Orthogonal Projection onto a Subspace Orthogonal Complement Pythongorean Theorem Cauchy-Schwarz Inequality Angle Between Two Vectors Gram-Schmidt Process QR Decomposition Orthongonal Transformation Transpose Symmetric and Skew Symmetric Matrices Matrix of an Orthogonal Projection Least-Squares Solution Inner Product Spaces Norm and Orthongonality Trace of a Matrix Orthogonal Projection of an Inner Product Space Fourier Analysis Determinants Sarrus s Rule Patterns Determinants and Gauss-Jordan Elimination Laplace Expansion Properties of the Determinant Eigenvalues and Eigenvectors Eigenvalues Characteristic Polynomial Algebraic Multiplicity Eigenvalues, Determinant, and Trace Eigenspace Eigenbasis Geometric Multiplicty Diagonalization Powers of a Diagonalizable Matrix Stable Equilibrium

4 9 Linear Differential Equations Exponential Growth and Decay Linear Dynamical Systems Continuous Dynamical Systems with Real Eigenvalues Continuous Dynamical Systems with Complex Eigenvalues Strategy for Solving Linear Differential Equations Eigenfunctions Characteristic Polynomial of a Linear Differential Operator Kernel of a Linear Differential Operator Characteristic Polynomial with Complex Solution First-Order Linear Differential Equations Strategy for Solving Linear Differential Equations Linearization of a Nonlinear System The Heat Equation The Wave Equation

5 1 Linear Equations 1.1 Standard Representation of a Vector v = 1.2 Reduced Row Echelon Form [ x y If a column contains a leading 1, then all ther other entires in that column are 0. If a row contains a leading 1, then each row above it contains a leading 1 further to the left. 1.3 Elementary Row Operations Divide a row by a nonzero scalar. Subtract a multiple of a row from another row. Swap two rows. 1.4 Rank of a Matrix The rank of a matrix A is the number of leading 1 s in rref(a). 1.5 Dot Product of Vectors 1.6 The Product A x A x = v w = v 1 w v n w n 1.7 Algebraic Rules for A x A( x y) = A x + A y A(k x) = k(a x) w 1. w n ] x = w 1 x. w n x 5

6 2 Linear Transformations 2.1 Linear Transformations A function T from R m to R n is called a linear transformation if there exists a matrix A such that T ( x) = A x A transformation is called linear if and only if T ( v + w) = T ( v) + T ( w) T (k v) = kt ( v) 2.2 Scaling Matrix A scaling matrix by k has the form: [ k 0 0 k 2.3 Orthogonal Projection onto a Line proj L ( x) = where w is a vector parallel to the line L. 2.4 Reflection Matrix A reflection matrix has the form: [ a b b a 2.5 Rotation Matrix ] ( ) x w w w ] w A rotation matrix has the form: [ cos θ sin θ sin θ cos θ ] 2.6 Shear Matrix A horizontal shear matrix has the form: [ 1 k 0 1 ] and a vertical shear matrix has the form: [ 1 0 k 1 ] 6

7 2.7 Matrix Multiplication The product of matrices BA is defined as the matrix of the linear transformation T ( x) = B(A x) = (BA) x. (AB)C = A(BC) A(B + C) = AB + AC (ka)b = A(kB) = k(ab) 2.8 Invertibility An n n matrix A is invertible if and only if: 1. rref(a) = I n 2. rank(a) = n or, more simply, if and only if det(a) Finding the Inverse 1. Form the n (2n) matrix [A I n ] 2. Compute rref[a I n ] 3. If rref[a I n ] is of the form [I n B] then A 1 = B 4. If rref[a I n ] is of another form, then A is not invertible 2.10 Properties of Invertible Matrices If an n n matrix A is invertible: The linear system A x = b has a unique solution x for all b in R n rref(a) = I n rank(a) = n im(a) = R n ker(a) = 0 The column vectors of A are linearly independent and form a basis of R n 7

8 3 Subspaces of R n and Their Dimensions 3.1 Image of a Function image(f) = {f(x) : x in X} = {b in Y : b = f(x), for some x in X} For a linear transformation T : The zero vector is in the image of T If v 1 and v 2 are in the image of T, then so is v 1 + v 2 If v is in the image of T, then so is k v 3.2 Span 3.3 Kernel span( v 1,, v n ) = {c 1 v c m v m : c 1 c n in R} The kernel of a linear transformation T is the solution set of the linear system: The zero vector is in the kernel of T A x = 0 If v 1 and v 2 are in the kerne; of T, then so is v 1 + v 2 If v is in the kernel of T, then so is k v 3.4 Subspaces of R n A subset W of the vector space R n is a linear subspace if and only if: 1. W contains the zero vector 2. W is closed under addition 3. W is closed under scalar multiplication The image and kernel of a linear transformation are linear subspaces. 3.5 Linear Independence 1. If a vector v i in v 1,, v n can be expressed as a linear combination of the vectors v 1,, v i 1, then v i is redundant. 2. The vectors v 1,, v n are linearly independent if no vector is redundant. 3. The vectors v 1,, v n form a basis of a subspace V if they span V and are linearly independent 8

9 3.6 Dimension The number of vectors in a subspace V is the dimension of V. 3.7 Rank-Nullity Theorem For an n m matrix A: 3.8 Coordinates dim(kera) + dim(ima) = m For a basis B = ( v 1,, v n ) of a subspace V, any vector x can be written as: x = c 1 v c n v n c 1,, c n are called the B-coordinates of x, with [ x] B = c 1. c n where S = [v 1 v n ] x = S[ x] B and [ x] B = S 1 x 3.9 Linearity of Coordinates [ x + y] B = [ x] B + [ y] B [k x] B = k[ x] B 3.10 Matrix of a Linear Transformation The matrix B that transforms [ x] B into [T ( x)] B is called the B-matrix of T : [T ( x)] B = B[ x] B, where B = [ [T ( v 1 )] B [T ( v n )] B ] 3.11 Similar Matrices Two matrices A and B are similar if and only if: AS = SB or B = S 1 AS 9

10 4 Linear Spaces 4.1 Linear Spaces A linear space V is a set with rules for addition and scalar multiplication that satisfies the following properties: 1. (f + g) + h = f + (g + h) 2. f + g = g + f 3. There exists a neutral element n in V such that f + n = f 4. For each f in V there exists a g in V such that f + g = 0 5. k(f + g) = kf + kg 6. (c + k)f = cf + kf 7. c(kf) = (ck)f 8. 1f = f 4.2 Finding a Basis of a Linear Space V 1. Find a typical element w of V in terms of arbitrary constants. 2. Express w as a linear combination of elements in V. 3. If these elements are linearly independent, they will for a basis of V. 4.3 Isomorphisms A linear transformation T is an isomorphism of T is invertible. A linear transformation T from V to W is an isomorphism if any only if ker(t ) = 0 and im(t ) = W. Coordinate changes are isomorphisms. If V is isomorphic to W, then dim(v ) = dim(w ). 10

11 5 Orthogonality and Least Squares 5.1 Orthongality and Length Two vectors c and w are orthogonal if v w = 0. The length of a vector v is v = v v. A vector u is a unit vector if its length is Orthonormal Vectors The vectors u 1,, u n are orthonormal if they are unit vectors orthogonal to each other. { 1 if i = j u i u j = 0 if i j 5.3 Orthogonal Projection onto a Subspace If V is a subspace with an orthonormal basis u 1,, u n : 5.4 Orthogonal Complement proj V ( x) = ( u 1 x) u ( u n x) u n The orthogonal complement V of a subspace V is the set of vectors x orthogonal to all vectors in V : V = { x in R n : v x = 0, for all v in V } V is a subspace of V V V = 0 dim(v ) + dim(v ) = n (V ) = V 5.5 Pythongorean Theorem 5.6 Cauchy-Schwarz Inequality x + y 2 = x 2 + y 2 x y x y 11

12 5.7 Angle Between Two Vectors 5.8 Gram-Schmidt Process For a basis v 1,, v n of a subspace V : where v i 5.9 QR Decomposition θ = arccos x y x y u 1 = 1 v 1 v 1,, u n = 1 v n = v i ( u 1 v i ) u 1 ( u i 1 v i ) v i 1 For an n m matrix M, M = QR, where Q is an n m matrix whose columns u 1,, u n are orthonormal and R has entries satisfying: r 1 1 = v 1, r jj = v j, r ij = u i v j for i < j 5.10 Orthongonal Transformation A linear transformation T is considered orthogonal if it preserves the length of vectors, such that T ( x) = x T is orthogonal if the vectors T ( e 1 ),, T ( e n ) form an orthonormal basis of R n. The matrix A is orthogonal if A T A = I n. The matrix A is orthogonal if A 1 = A T. A matrix A is orthogonal if its columns form an orthonormal basis of R n. The product AB of two orthogonal matrices A and B is orthogonal. The inverse A 1 of an orthogonal matrix A is orthogonal Transpose The transpose A T of an m n matrix A is the n m matrix whose ijth entry is the jith entry of A. (AB) T = B T A T (A T ) 1 = (A 1 ) T rank(a) = rank(a T ). 12

13 5.12 Symmetric and Skew Symmetric Matrices An n n matrix A is symmetric if A T = A. An n n matrix A is skew-symmetric if A T = A Matrix of an Orthogonal Projection The orthongonal projection onto a subspace V with an orthonormal basis u 1,, u n is QQ T, where Q = [ u 1 u n ] or equivalently, A(A T A) 1 A T where A = [ v 1 v n ] 5.14 Least-Squares Solution The unique least-squares solution of a linear system A x = b where ker(a) = 0 is x = (A T A) 1 A T b 5.15 Inner Product Spaces The inner product of a linear space V, denoted f, g, has the following properties: f, g = g, f f + h, g = f, g + h, g cf, g = c f, g f, f > Norm and Orthongonality The norm of an element f of an inner product space is f = f, f Two elements f and g of an inner product space are orthongonal if f, g = Trace of a Matrix The trace tr(a) of a matrix A is the sum of its diagonal entries. 13

14 5.18 Orthogonal Projection of an Inner Product Space If g 1,, g n is an orthonormal basis of a subspace W of an inner product space V : 5.19 Fourier Analysis proj W f = g 1, f g g n, f g n f n (t) = a b 1 sin(t) + c 1 cos(t) + + b n sin(nt) + c n cos(nt) where a 0 = f, 1 = 1 2 π 2 b k = f, sin(kt) = 1 π c k = f, cos(kt) = 1 π π π π π π π f(t) dt f(t) sin(kt) dt f(t) cos(kt) dt 6 Determinants 6.1 Sarrus s Rule For an n n matrix A, write the first n 1 columns to the right of A, then multiply along the diagonal to get 2n products. Subtract the first n products, then add the second n products to get the determinant. 6.2 Patterns A pattern of an n n matrix A is a way to choose n entries of A such that each entry is in a unique row and column. The product of a pattern is designated P. Two entries in a pattern are an inversion if one is located above and to the right of the other. The signature of a pattern is defined as sgn P = ( 1)(inversions in P) det(a) = (sgn P )(prod P ) 14

15 6.3 Determinants and Gauss-Jordan Elimination If B is an n n matrix obtained from applying an elementary row operation on an n n matrix A: If B is obtained by row division: det(b) = 1 k det(a) If B is obtained by row swap: det(b) = det(a) If B is obtained by row addition: det(b) = det(a) 6.4 Laplace Expansion Expansion down the jth column: det(a) = n ( 1) i+j a ij det(a ij ) i=1 Expansion along the ith row: det(a) = n ( 1) i+j a ij det(a ij ) j=1 6.5 Properties of the Determinant det(a T ) = det(a) det(ab) = det(a)det(b) If A and B are similar, then det(a) = det(b) det(a 1 ) = 1 det(a) = det(a) 1 7 Eigenvalues and Eigenvectors 7.1 Eigenvalues λ is an eigenvalues of an n n matrix A if and only if 7.2 Characteristic Polynomial det(a λi n ) = 0 det(a λi n ) = ( 1) n λ n + ( 1) n 1 tr(a)λ n det(a) 15

16 7.3 Algebraic Multiplicity An eigenvalue λ has algebraic multiplicty k if it is a root of multiplicity k of the characteristic polynomial. 7.4 Eigenvalues, Determinant, and Trace For an n n matrix A det(a) = λ 1 λ n = tr(a) = λ λ n = n k=1 λ k n k=1 λ k 7.5 Eigenspace E λ = ker(a λi n ) = { v in R n : A v = λ v} 7.6 Eigenbasis An eigenbasis for an n n matrix A conists of the eigenvectors of A and forms a basis for R n. 7.7 Geometric Multiplicty The dimension of the eignspace E λ is the geometric multiplicty of the eigenvalue λ. 7.8 Diagonalization 1. Find the eigenvalues and corresponding eigenvectors of the matrix A. 2. Let S be the eigenbasis for A and D be a matrix with the eigenvalues of A along the diagonal. 3. D = S 1 AS and A = SDS Powers of a Diagonalizable Matrix If A = SDS 1, then A t = SD t S 1 16

17 7.10 Stable Equilibrium 0 is an asympototically stable equilibrium for the system x(t + 1) = A x(t) if lim x(t) = lim t t At = 0 9 Linear Differential Equations 9.1 Exponential Growth and Decay 9.2 Linear Dynamical Systems Discrete model: x(t + 1) = B x(t) Continuous model: d x dt = A x dx dt = kx, x(t) = ekt x Continuous Dynamical Systems with Real Eigenvalues d x dt = A x, x(t) = c 1e λ 1t v c n e λnt v n where v 1, v n forms a real eigenbasis of A with eignvalues λ 1,, λ n 9.4 Continuous Dynamical Systems with Complex Eigenvalues d x dt = A x, [ cos(qt) sin(qt) x(t) = ept S sin(qt) cos(qt) where p ± q are eigenvalues with eigenvectors v ± w and S = [ w v]. ] S 1 x Strategy for Solving Linear Differential Equations To solve an nth order linear differential equation with the form T (f) = g: 1. Find a basis f 1,, f n of ker(t ). 2. Find a particular solution f p. 3. The solutions f are in the form f = c 1 f c n f n + f p. 17

18 9.6 Eigenfunctions A smooth function F is an eigenfunction of T if T (f) = λf 9.7 Characteristic Polynomial of a Linear Differential Operator For T (f) = f (n) + a n 1 f (n 1) + + a 1 f + a 0 f, the characteristic polynomial is defined as: p T (λ) = λ n + a n 1 λ n a 1 λ + a 0 If e λt is an eigenfunction of T with eigenvalue p T (λ): T (e λt ) = p T (λ)e λt 9.8 Kernel of a Linear Differential Operator If T is a linear differential operator with characteristic polynomial p T (λ) with roots λ 1,, λ n, then the kernel of T is formed by e λ 1t, e λ 2t,, e λnt 9.9 Characteristic Polynomial with Complex Solution If the zeros of p T (λ) are p ± q, then the solutions to its differential equation are of the form f(t) = e pt (c 1 cos(qt) + c 2 sin(qt)) 9.10 First-Order Linear Differential Equations A differential equation of the form f (t) af(t) = g(t) has a solution of the form f(t) = e at e at g(t) dt 9.11 Strategy for Solving Linear Differential Equations To solve the nth order linear differential equation T (f) = g: 1. Find n linearly independent solutions of T (f) = 0. Write the characteristic polynomial p T (λ) of T by replacing f (k) with λ k Find the solutions λ 1,, λ n of the equation p T (λ) = 0. If λ is a solution of p T (λ) = 0, then e λt is a solution of T (f) = 0. 18

19 If λ is a solution of p t (λ) = 0 with multiplicity m, then e λt, te λt, t 2 e λt,, t m 1 e λt are the solutions of T (f) = 0. If p ± q are complex solutions of p T (λ) = 0, then e pt cos(qt) and e pt sin(qt) are real solutions of T (f) = If T (f) is inhomogenous, find one particular solution f p of the equation T (f) = g. If g is of the form g(t) = A cos(ωt) + B sin(ωt), g(t) = A cos(ωt), or g(t) = A sin(ωt), look for a particular solution of the form f p (t) = P cos(ωt) + Q sin(ωt). If g is of the form g(t) = a 0 + a 1 t + a n t n, look for a particular solution of the form f p (t) = c 0 + c 1 t + c n t n If g is constant, look for a particular solution of the form f p (t) = c. If g is of the form g(t) = f (t) af(t), use the formula f(t) = e at e at g(t) dt. 3. The solutions of T (f) = g are of the form f(t) = c 1 f 1 (t) + c 2 f 2 (t) + + c n f n (t) + f p (t) where f 1,, f n are the solutions from Step 1 and f p is the solution from Step Linearization of a Nonlinear System [ f(x, y) If (a, b) is an equilibrium point of the system g(x, y) 0, then the system is approximated near (a, b) by the Jacobian matrix: ] [ ] f f [ ] (a, b) (a, b) x y u = g (a, b) (a, b) v [ du dt dv dt 9.13 The Heat Equation g x has solutions of the form f(x, t) = b n sin(nx)e n2 µt n= The Wave Equation y f t (x, t) = µf xx (x, t) where b n = 2 π f tt (x, t) = c 2 f xx (x, t) ], such that f(a, b) = 0 and g(a, b) = π 0 f(x, 0) sin(nx) dx has solutions of the form f(x, t) = a n sin(nx) cos(nct) + b n sin(nx) sin(nct) nc n=1 where a n = 2 π f(x, 0) sin(nx) dx and b π 0 n = 2 π f π 0 t(x, 0) sin(nx) dx. 19

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