MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones will not be allowed during the test 1 Vectors in R 2 and R 3 2 Systems of linear equations 3 Elementary row operations 4 Gaussian elimination 5 Matrices 6 Matrix addition and scalar multiplication 7 Matrix multiplication 8 Inverse matrices 9 Elementary matrices 10 Determinants 11 Vectors in R n 12 Subspaces of R n 13 Linear combinations in R n 14 Linear independence in R n 15 Bases and dimension in R n Topics 1
Practice Problems from Edwards & Penney, Diff Eq & Lin Alg, 3rd ed Section 31: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 31, 32, 33, 34 Section 32: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 Section 34: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 29, 30, 33, 34, 35, 36, 37, 38 Section 35: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 32, 34 Section 36: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 50, 54 Section 41: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36 Section 42: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 Section 43: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 Section 44: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 2
Some Important Concepts and Formulas 1 Vectors in R 2 and R 3 : Addition and Scalar Multiplication Let a,b,c R 2 or a,b,c R 3, and let s,t R Let 0 be the zero vector 1 a + b = b + a (Commutative Law) 2 a + (b + c) = (a + b) + c (Associative Law) 3 a + 0 = a and 0 + a = a (Identity Law) 4 a + ( a) = 0 and ( a) + a = 0 (Inverses Law) 5 s(a + b) = sa + sb (Distributive Law) 6 (s +t)a = sa +ta (Distributive Law) 7 s(ta) = (st)a 8 1a = a 2 Vectors in R 3 : Coordinates 1 The set R 3, thought of as vectors, is the set of all column vectors of the form a = with entries in R [ 10 [ 01 [ 00 [ a1 2 Let i =, and j = and k = Then 2 = a 0 0 1 a 1 i + a 2 j + a 3 k 3 [ a1 3 The length (also known as norm) of a vector a = 2 in R 3 is a 3 a = 4 The vector 0 is the zero vector in R 3 (a 1 ) 2 + (a 2 ) 2 + (a 3 ) 2 [ a1 2 a 3 3 Vectors in R 2 and R 3 : Unit Vectors If a is a vector in R 2 or R 3, and if a 0, the unit vector in the direction of a is u = 1 a a 3
4 Vectors in R 3 : Addition and Scalar Multiplication via Coordinates [ a1 [ b1 Let a = 2 and b =, and let c R a 3 1 a + b = 2 a b = 3 a = 4 ca = [ a1 +b 1 a 2 +b 2 a 3 +b 3 [ a1 b 1 a 2 b 2 a 3 b 3 [ a1 a 2 a 3 [ ca1 ca 2 ca 3 b 2 b 3 5 Systems of Linear Equations 1 Let m,n N A system of m linear equations in n unknowns is a system of equations with unknowns x 1,x 2,,x n that can be written in the form a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m, for some a 11,a 12,,a mn R and b 1,b 2,,b m R 2 A system of m linear equations in n unknowns is homogeneous if it has the form a 11 x 1 + a 12 x 2 + + a 1n x n = 0 a 21 x 1 + a 22 x 2 + + a 2n x n = 0 a m1 x 1 + a m2 x 2 + + a mn x n = 0 6 Systematic Solution of a System of Linear Equations: Preliminary 1 Let m,n N An m n matrix with entries in R is a rectangular array of real numbers with m rows and n columns 4
2 Given a system of m linear equations in n unknowns of the form a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m, the coefficient matrix of the system is a 11 a 12 a 13 a 1n a 21 a 22 a 23 a 2n A =, a m1 a m2 a m3 a mn and the augmented coefficient matrix is a 11 a 12 a 13 a 1n b 1 [ a 21 a 22 a 23 a 2n b 2 A b = a m1 a m2 a m3 a mn b m 3 The augmented coefficient contains all the information needed to solve the system of linear equations 7 Echelon Form 1 In a matrix, a row is non-zero if it has a least one non-zero entry 2 In a matrix, the leading entry of a non-zero row is the first non-zero entry in a row (reading from the left) 3 A matrix is in echelon form if the following two properties hold: (a) Each row that is all zeros is below every non-zero row (b) The leading entry of each non-zero row lies to the right of the leading entry of the preceding row (if there is a preceding row) 4 A matrix is in partially reduced echelon form if the following three properties hold: (a) Each row that is all zeros is below every non-zero row (b) The leading entry of each non-zero row lies to the right of the leading entry of the preceding row (if there is a preceding row) (c) The leading entry of each non-zero row is 1 5
5 A matrix is in reduced echelon form if the following four properties hold: (a) Each row that is all zeros is below every non-zero row (b) The leading entry of each non-zero row lies to the right of the leading entry of the preceding row (if there is a preceding row) (c) The leading entry of each non-zero row is 1 (d) The leading entry of each non-zero row is the only non-zero entry in its column 8 Elementary Row Operations There are three elementary row operations that are performed on matrices 1 Multiplying a row by a non-zero real number 2 Interchanging two rows 3 Adding a multiple of one row to another row 9 Systematic Solution of a System of Linear Equations: Part 1 Gaussian Elimination To solve a system of linear equations, start with the following steps 1 Form the augmented coefficient matrix [ A b 2 The first part of the process, called Gaussian elimination (or Gauss-Jordan elimination if the steps in parentheses are used), occurs in the the augmented coefficient matrix (a) Find the first column of the augmented coefficient matrix (reading from the left) that has a non-zero entry (b) If the top entry in this column is zero, place a non-zero element there by interchanging appropriate rows (c) Make this non-zero entry into 1 by multiplying the row containing the entry by an appropriate number (d) Make all the entries below this non-zero entry into zero by adding appropriate multiples of the row containing this non-zero entry to these other rows (e) (For Gauss-Jordan elimination, make all the entries above this non-zero entry into zero by adding appropriate multiples of the top row to these other rows) (f) The top row is now complete, and will not be modified further Continue the above process to the rows below the top row, one row at a time, starting from the top 6
(g) Keep going until partially reduced echelon form (or reduced echelon form for Gauss-Jordan elimination) is achieved 3 If there is a row in the matrix in partially reduced echelon form (or reduced echelon form) that has all zeros except for the last entry, which is not zero, then there is no solution Otherwise there is a solution (one or infinitely many, to be determined) 4 If there is a solution, convert the matrix in partially reduced echelon form (or reduced echelon form) back into equations 5 The variables in the new equations that correspond to leading entries are leading variables The other variables are free variables 6 If there are no free variables, then there is one solution If there are free variables, there are infinitely many solutions, with as many parameters as there are free variables 10 Systematic Solution of a System of Linear Equations: Part 2 Back Substitution To solve a system of linear equations, finish with the following steps 1 The next part of the process, called back substitution, occurs with these new equations (a) Set each free variable equal to a parameter (each free variable is a different parameter) (b) Solve the final non-zero equation for its lead variable (in terms of the parameters, if there are any in that equation) (c) Substitute that solution into the equation above it, and solve for its lead variable (d) Keep going upwards until all free variables have been found 11 Matrices 1 Let m,n N The set M m n (R) is the set of all m n matrices with entries in R 2 Let A M m n (R) Then A = [ a11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn 3 The transpose of A is the n m matrix A T = 4 The matrix O M m n (R) is the zero matrix [ a11 a 21 a m1 a 12 a 22 a m2 a 1n a 2n a mn 5 The matrix I n M n n (R) is the identity matrix I n = 7 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
12 Matrices: Addition and Scalar Multiplication Let A = 1 A + B = [ a11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn 2 A B = 3 A = 4 ca = and B = a 11 +b 11 a 12 +b 12 a 1n +b 1n a 21 +b 21 a 22 +b 22 a 2n +b 2n a m1 +b m1 a m2 +b m2 a mn +b mn a 11 b 11 a 12 b 12 a 1n b 1n a 21 b 21 a 22 b 22 a 2n b 2n a m1 b m1 a m2 b m2 a mn b mn a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn [ ca11 ca 12 ca 1n ca 21 ca 22 ca 2n ca m1 ca m2 ca mn b 11 b 12 b 1n b 21 b 22 b 2n b m1 b m2 b mn, and let c R 13 Matrices: Multiplication Row times Column b 1 [ b 2 a1 a 2 a n = a 1b 1 + a 2 b 2 + a n b n b n General If A is an m p matrix and B is a p n matrix, then AB is an m n matrix obtained by multiplying each row in A by each column in B 8
14 Matrices and Systems of Linear Equations Let m,n N A system of m linear equations in n unknowns can be written in the form a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m Define the matrices A M m n (R), and B R m and X = R n by a 11 a 12 a 13 a 1n x 1 b 1 a 21 a 22 a 23 a 2n A = and X = x 2 and B = b 2 a m1 a m2 a m3 a mn x n b m Then the system of linear equations is equivalent to the matrix equation AX = B Solving for the vector X is equivalent to solving for the unknowns x 1,,x n 15 Matrices: Properties of Addition and Scalar Multiplication Let m,n N Let A,B,C M m n (R), and let s,t R Let O M m n (R) be the zero matrix 1 A + B = B + A (Commutative Law) 2 A + (B +C) = (A + B) +C (Associative Law) 3 A + O = A and A + O = A (Identity Law) 4 A + ( A) = O and ( A) + A = O (Inverses Law) 5 s(a + B) = sa + sb (Distributive Law) 6 (s +t)a = sa +ta (Distributive Law) 7 s(ta) = (st)a 8 1A = A 9
16 Matrices: Properties of Multiplication Let m,n, p,q N Let A,P M m n (R), let B,Q M n p (R) and let C M p q (R) 1 A(BC) = (AB)C 2 AI n = A and I m A = A 3 A(B + Q) = AB + AQ and (A + P)B = AB + PB 17 Matrices: Properties of Powers Let n N Let A M n n (R), and let r,s N {0} 1 A r A s = A r+s 2 (A r ) s = A rs 18 Matrices: Properties of the Transpose Let m,n N Let A,B M m n (R), and let x R 1 (A + B) T = A T + B T 2 (ca) T = ca T 3 (A T ) T = A 4 (I n ) T = I n 5 (AB) T = B T A T 6 A is invertible if and only if A T is invertible; if A is invertible, then (A T ) 1 = (A 1 ) T 19 Matrices: Inverses Let n N Let A,B M n n (R) 1 The matrix A is invertible if there is some C M n n (R) such that CA = I n and AC = I n Such a matrix C, if it exists, is unique, and is called the inverse of A, and is denoted A 1 2 If A is invertible, then A 1 is invertible and (A 1 ) 1 = A 3 If A and B are invertible, then AB is invertible and (AB) 1 = B 1 A 1 10
20 Matrices: Inverses of 2 2 Matrices Let A = [ a b c d 1 The matrix A is invertible if and only if ad bc 0 2 If A is invertible, then A 1 = 1 ad bc [ d b c a 21 Elementary Row Operations and Elementary Matrices Elementary Row Operations There are three elementary row operations that are performed on matrices 1 Multiplying a row by a non-zero real number 2 Interchanging two rows 3 Adding a multiple of one row to another row Row Equivalence Two matrices are row equivalent if one matrix can be obtained from the other by a finite number of elementary row operations Elementary Matrices An elementary matrix is the result of doing a single elementary row operation to the identity matrix 22 Finding Matrix Inverses Let n N Let A M n n (R) 1 The matrix A is invertible if and only if A is row equivalent to I n 2 Suppose that A is invertible The augmented matrix [A I n is row equivalent to [I n A 1 To find A 1, start with [A I n, and perform elementary row operations on it until the left side is I n 11
23 Application of Invertible Matrices Let n N Let A M n n (R) The following are equivalent (a) A is invertible (b) A is row equivalent to I n (c) AX = 0 has only the trivial (all zero) solution (d) For every n-vector B, the system AX = B has a unique solution (e) For every n-vector B, the system AX = B is consistent 24 Determinants of 2 2 Matrices det [ a b c d = ad bc 25 Minors and Cofactors Let n N Let A M n n (R) Let i, j {1,,n} 1 The i j th minor of A, denoted M i j, is the determinant of the (n 1) (n 1) submatrix of A obtained by deleting the i th row and j th column of A 2 The i j th cofactor of A, denoted A i j, is ( 1) i+ j M i j 3 The cofactor matrix of A, denoted cofa, is the matrix [ A i j 26 Determinants of n n Matrices Let n N Let A M n n (R) The determinant of A, denoted deta, can be obtained by expanding along any row or any column and using minors or cofactors The same result will be obtained no matter which row or column is chosen 1 Let i {1,,n} Then expanding along the i th row yields deta = ( 1) i+1 a i1 M i1 + ( 1) i+2 a i2 M i2 + + ( 1) i+n a in M in = a i1 A i1 + a i2 A i2 + + a in A in 2 Let j {1,,n} Then expanding along the j th column yields deta = ( 1) 1+ j a 1 j M 1 j + ( 1) 2+ j a 2 j M 2 j + + ( 1) n+ j a n j M n j = a 1 j A 1 j + a 2 j A 2 j + + a n j A n j 12
27 Properties of Determinants Let n N Let A,B,C M n n (R), and let k R 1 If B is obtained from A by multiplying a row (or column) of by k, then detb = k deta 2 If B is obtained from A by interchanging two rows (or two columns), then detb = deta 3 If two rows (or two columns) of A are equal, then deta = 0 4 If B is obtained from A by adding a multiple of one row (or column, respectively) to another row (or column, respectively), then detb = deta 5 If A is a triangular matrix, then deta is the product of the diagonal elements of A 6 det(a T ) = deta 7 det(ab) = deta detb 28 Determinants and Inverse Matrices Let n N Let A M n n (R) 1 The matrix A is invertible if and only if deta 0 2 If A is invertible, then 3 If A is invertible, then det(a 1 ) = 1 deta A 1 = 1 deta (cofa)t 13
29 Determinants and Solutions of Systems of Linear Equations Determinants are useful for solutions of systems of linear equations when the number of equations is the same as the number of unknowns [ x1 Let n N Let A M n n (R), let B R n x 2, and let X = x n 1 If deta 0, then AX = B has a unique solution, which is X = A 1 B 2 If deta 0, then AX = 0 has a unique solution, which is X = 0 3 If deta = 0, then AX = B either has no solution or it has infinitely many solutions 4 If deta = 0, then AX = 0 has infinitely many solutions 30 Cramer s Rule Let n N Let A M n n (R), let B R n, and let X = [ x1 x 2 x n 1 For each i {1,,n}, let R i bes the matrix obtained by replacing the i th column of A with B 2 Suppose that A is invertible Then the system of linear equations AX = B has a unique solution, which is given by x i = detr i deta for all i {1,,n} 31 Vectors in R n Let n N 1 The set R n is the set of all column vectors of length n with entries in R [ v1 2 Let v R n v 2 Then v = v n 3 The vector 0 R n is the zero vector 4 The length (also known as norm) of a vector v = v = (v 1 ) 2 + (v 2 ) 2 + + (v n ) 2 [ v1 v 2 in R n, denoted v, is defined by v n 14
5 If v is a vector in R n, and if v 0, the unit vector in the direction of v is u = 1 v v 32 Vectors in R n : Properties of Length Let n N Let v R n, and let c R 1 v 0 2 v = 0 if and only if v = 0 3 cv = c v 33 Vectors in R n : Addition and Scalar Multiplication [ x1 [ y1 x 2 y 2 Let n N Let x = and y =, and let c R x n y n 1 x + y = 2 x y = 3 x = 4 cx = x 1 +y 1 x 2 +y 2 x n +y n x 1 y 1 x 2 y 2 x 1 x 2 x n [ cx1 cx 2 cx n x n y n 34 Vectors in R n : Properties of Addition and Scalar Multiplication Let n N Let u,v,w R n, and let s,t R 1 u + v = v + u (Commutative Law) 2 u + (v + w) = (u + v) + w (Associative Law) 3 u + 0 = u and 0 + u = u (Identity Law) 4 u + ( u) = 0 and ( u) + u = 0 (Inverses Law) 15
5 s(u + v) = su + sv (Distributive Law) 6 (s +t)u = su +tu (Distributive Law) 7 s(tu) = (st)u 8 1u = u 35 Subspaces Let n N Let W R n be a non-empty subset 1 The subset W is closed under addition if u + v W for all u,v W 2 The subset W is closed under scalar multiplication if sv W for all v W and all s R 3 The subset W is a subspace of R n if it is closed under addition and scalar multiplication 36 Linear Combinations Let n N Let v 1,,v k R n A linear combination of vectors of v 1,,v k is any vector of the form a 1 v 1 + a 2 v 2 + + a k v k for some a 1,a 2,,a k R 37 Span Let n N Let v 1,,v k R n 1 The span of v 1,,v k, denoted span{v 1,,v k }, is the set of all linear combinations of the vectors v 1,,v k 2 {v 1,,v k } span{v 1,,v k } 3 The subset span{v 1,,v k } is a subspace of R n 4 The vectors v 1,,v k span R n if span{v 1,,v k } = R n 16
38 Linear Independence Let n N Let v 1,,v k R n 1 The vectors v 1,,v k are linearly dependent if there are a 1,a 2,a k R that are not all 0, such that a 1 v 1 + + a k v k = 0 2 The vectors v 1,,v k are linearly independent if they are not linearly dependent 39 Strategy for Proving Linear Independence Let n N Let v 1,,v k R n The standard strategy for showing that v 1,,v k are linearly independent is as follows: Suppose that a 1 v 1 + + a k v k = 0 for some a 1,a 2,a k R (argumentation) Then a 1 = 0,, a n = 0 Hence v 1,,v k are linearly independent 40 Bases of Subspaces of R n Let n N Let W R n be a subspace Let v 1,,v k W 1 The vectors v 1,,v k are a basis for W if they are linearly independent and they span W 2 If v 1,,v k are a basis for W, then every vector v W can be expressed uniquely as a linear combination of v 1,,v k 3 If every vector v W can be expressed uniquely as a linear combination of v 1,,v k, then v 1,,v k are a basis for W 4 The subspace W has a basis, and all bases of W have the same number of vectors 41 Dimension of Subspaces of R n Let n N Let W R n be a subspace The dimension of W, denoted dimw, is the number of vectors in any basis for W 17
42 Properties of Linearly Independent and Spanning Sets in Subspaces of R n Let n N Let W R n be a subspace Let m = dimw Let v 1,,v k W 1 If v 1,,v k span W, then k m 2 If v 1,,v k span W and k = m, then v 1,,v n are a basis for W 3 If v 1,,v k are linearly independent, then k m 4 If v 1,,v k are linearly independent and k = m, then v 1,,v n are a basis for W 5 If v 1,,v k is spans W, then it contains a basis for W 6 If v 1,,v k is linearly independent, then it can be extended to a basis for W 43 Properties of Linearly Independent and Spanning Sets in R n Let n N Let v 1,,v k R n 1 If v 1,,v k span R n, then k n 2 If v 1,,v k span R n and k = n, then v 1,,v n are a basis for R n 3 If v 1,,v k are linearly independent, then k n 4 If v 1,,v k are linearly independent and k = n, then v 1,,v n are a basis for R n 5 If v 1,,v k is spans R n, then it contains a basis for R n 6 If v 1,,v k is linearly independent, then it can be extended to a basis for R n 18