a 1n a 2n. a mn, a n = a 11 a 12 a 1j a 1n a 21 a 22 a 2j a m1 a m2 a mj a mn a 11 v 1 + a 12 v a 1n v n a 21 v 1 + a 22 v a 2n v n

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1 Let a 1,, a n R m and v R n a 11 a 1 = a 21, a 2 = a m1 a 12 a 22 a m2 Matrix-vector product, a j = a 1j a 2j a mj Let A be the m n matrix whose j-th column is a j : Define A = [ a 1 a 2 a j a n ] = A v = v 1 a 1 + v 2 a 2 + v n a n =, a n = a 1n a 2n a mn, and v = a 11 a 12 a 1j a 1n a 21 a 22 a 2j a 2n a m1 a m2 a mj a mn m n a 11 v 1 + a 12 v a 1n v n a 21 v 1 + a 22 v a 2n v n a i1 v 1 + a i2 v a in v n a m1 v 1 + a i2 v a mn v n v 1 v 2 v n

2 Matrix-vector product:example and properties Example = Key property of Matrix-vector product: Let A be a m n matrix, let u, v be column vectors of length n and let c, d R Then A(c u + d v) =ca u + da v The verification is routine

3 Consider again the system: Vector and matrix form of system of linear equations x 1 = 3 2x 3 + 8x 4 6x 5 = 2 x 1 + x 2 2x 4 = 9 3x 2 2x 3 14x 4 + 5x 5 = 16 1 Rewrite in vector form: x x x x x 5 6 = Rewrite in matrix form: x x x 3 x = x

4 Vector and matrix form of system of linear equations: General form Consider a system of m linear equations in n variables x 1,, x n : a 11 x 1 + a 12 x a 1j x j + + a 1n x n = b 1 a 21 x 1 + a 22 x a 2j x j + + a 2n x n = b 2 a i1 x 1 + a i2 x a ij x j + + a in x n = b i a m1 x 1 + a m2 x a mj x j + + a mn x n = b m Vector form: x 1 a 1 + x 2 a x n a n = b where a1 = a 11 a 21 a m1, a 2 = a 12 a 22 a m2, a n = a 1n a 2n a mn, and b = b 1 b 2 b m m 1 Matrix form: A x = b where A = a 11 a 12 a 1j a 1n a 21 a 22 a 2j a 2n a m1 a m2 a mj a mn m n and x = x 1 x 2 x n n 1

5 Consistency of linear system in terms of linear combo and span Example Let a 1, a 2, a 3, b be the following vectors in R 3 : [ ] [ ] [ ] [ ] a 1 =, a 2 =, a 3 =, b = Question: Is b alinearcombinationof a1, a 2, a 3? Rephrase Q Does b belong to span{ a1, a 2, a 3 }? Let A be the 3 3 matrix whose first, second and third columns are a 1, a 2, a 3 A = [ a 1 a 2 a 3 ] Rephrase Q Again: Does the system equation x 1 a 1 + x 2 a 2 + x 3 a 3 = b (or A x = b)havea solution? In other words, is the system consistent? Solution Recall that we solved the linear system A x = b by row reduction last week The system has no solution, ie is inconsistent So b is not a linear combination of a 1, a 2, a 3 What is going on Geometrically a 1 and a 2 span a plane P in R 3 since a 1, a 2 are not scalar multiple of each other Observe: a 3 = a 1 + a 2 So a 3 is a linear combo of a 1 and a 2 In other words, a 3 lies in the plane P So a 1, a 2, a 3 also span the plane P The vector b does not lie on this plane P So b cannot be written as a linear combination of a 1, a 2, a 3

6 More on Span Question Do the following vectors span R 4? Theorem Let a 1,, a n R m Let A = [ a 1 a 2 a n ] The following are equivalent (TFAE): span{ a 1,, a n } = R m Every vector b R m can be written as a linear combo of a 1,, a n The vector equation x 1 a x n a n = b has a solution for all b R n The equation A x = b has a solution for all b R m The reduced echelon form of A has a pivot in every row

7 Homogeneous systems AlinearsystemofequationsoftheformA x = is called a homogeneous system Important property of homogeneous system If u and v are solutions of A x = and c, d R, then (c u + d v) is also a solution of A x = Example Solve thehomogeneous linear system A x = where A = We do not bother with the augmented matrix because the last column of the Augmented matrix is going to be all zeroes From earlier computation, recall the reduced row echelon form of A: Note the the 4-th column ha no pivot, so x 4 is free We write x 4 = t where t is any real number General solution Another example: x 1 x 2 x 3 x = 2t 4t 4 t = t x 5

8 Compare solutions of homogeneous and nonhomogeneous systems Consider a non-homogeneous system A x = b Suppose A x = b is consistent Fix a solution v of A x = blet v be any solution of A x = blet u = v v Observe: A u = A v A v = b b = So u is a solution of A x = Thusany solution of A x = b can be written in the form v = v + u where u is a solution of A x = Conversely, verify that adding a solution of A x = to a solution of A x = b again gives a solution of A x = b Summarizing we get: Theorem Let v be a particular solution of A x = bthen the set of are solutions of A x = b are ( v + u) where u varies over the set of solutions of A x = Example: Consider again the system A x = b where A = 11 2 and 2 b = Recall x 4 was a free variable We write x 4 = t So general solution of A x = b has the form t 1 4t t = t The first term on the right side is a particular solution of A x = b and the second term gives the general solution of A x =

9 Linear independence Example Definition Let { a 1, a 2,, a n } be an indexed set of vectors in R n Alinearcombination c 1 a 1 + c 2 a c n a n is called the trivial linear combination of { a 1,, a n } if c j = for all j = 1, 2,, n Otherwise, we say c 1 a 1 + c 2 a c n a n is a non-trivial linear combination of { a 1, a 2,, a n } We say { a 1, a 2,, a n } is linearly dependent if anontriviallinearcombinationof { a 1, a 2,, a n } is equal to the zero vector We say that { a 1, a 2,, a n } is linearly independent, iftheyarenotlinearly dependent

10 Example Are the following vectors a 1, a 2, a 3, a 4 linearly dependent or independent? If they are linearly dependent, then find a nontrivial dependence relation among them a 1 = 1, a 2 = 1 a 3 = a 4 = Solution: Let B be the matrix B = [ a 1 a 2 a 3 a 4 ] The vectors are linearly dependent if x 1 a 1 + x 2 a 2 + x 3 a 3 + x 4 a 4 = or B x = has a non-trivial solution Compute the row echelon form of B by row reduction: Note that there is a pivot free column So B x = has infinitely many solutions In particular, B x = has a nontrivial solution So a 1, a 2, a 3, a 4 are linearly dependent To find a dependence relation, we find a nontrivial solution Note that x 4 is a free variable Choose x 4 = 1 Thisgivesthesolution x 4 = 1, x 3 = 4, x 2 = 2, x 1 = So a nontrivial dependence relation is 2 a 2 4 a 3 + a 4 =

11 Detecting independence Let { a 1, a 2,, a n } be an indexed set of column vectors in R m WriteA = [ a 1 a 2 a n ]TFAE: The set { a 1, a 2,, a n } is linearly indep The only solution of A x = is x = The row echelon form of A has a pivot in each column Alistofm linearly indep vectors in R m are: 1 e 1 =, e 2 = 1, e m = 1 These are called standard basis vectors of R m Detecting dependence If an indexed set of vectors contain the vector, then it is linearly dependent If an indexed set of vectors is linearly dependent, then so is any larger collection of vectors (m + 1) or more vectors in R m must be dependent Reason: A has more columns than rows, so there must be a pivot free column in A red An indexed set { a 1, a 2,, a n } of vectors in R m is linearly dependent if and only if some a j can be written as a linear combination of a 1,, a j 1 or if a 1 =

12 Application: Balancing chemical equation H 3 O + CaCO 3 H 2 O + Ca + CO 2 Suppose (x 1 )H 3 O+(x 2 )CaCO 3 (x 3 )H 2 O+(x 4 )Ca+(x 5 )CO 2 Count H atoms : 3x 1 + x 2 = 2x 3 + x 4 + x 5 Soution by inspection: Balanced equation: x 1 x 2 x 3 x = 4 x Count O atoms : x 1 + 3x 2 = x 3 + x 4 + 2x 5 (2)H 3 O + CaCO 3 (3)H 2 O + Ca + CO 2 Similarly for Ca and C This gives: 3 2 x x x x = 4 x 5 Find a nontrivial solution with non-negative integer entries that have no common factor

13 Application: Network flow Exercise 12 (Section 16): (See figure) To find the general flow pattern write down the linear system of equations obtained by equating the inflow and outflow at each node Node A: x 1 + x 4 = x 2 Node B: x 2 = x Node C: x = x 4 Rearranging, we get the system: [ ] x x x = 3 x 4 [ ] 1 8 Find a solution with non-negative entries

14 Linear Transformation Example Definition AfunctionT : R n R m is called a linear transformation if Scaling T( u + v) =T( u)+t( v) and T(c v) =ct( v) for all u, v R n and for all c R First examples in geometric terms Shear Rotation about an axis going through origin Inclusion and Projection Reflection across mirror through origin Theorem Composition of Linear maps are linear (This is easy to verify)

15 Representing linear maps algebraically: Example Consider the linear map T : R 2 R 2 that rotates each vector [ anticlockwise [ by 9 degrees Recall that we write 1 e 1 = ] and e 2 = 1] Note that T( e 1 )= [ 1 ] and T( e 2 )= [ ] 1 Key observation: Because of linearity, T( e 1 ) and T( e 2 ) determine T Forexample: [ ] 32 T( )=T(3 e e 2 ) = 3T( e 1 )+2T( e 2 ) [ ] [ ] 1 1 = [ ] 1 The matrix A = [T( e 1 ) T( e 2 )] = 1 is called the matrix of the linear map T : R 2 R 2 (with respect to the standard basis) The general statements Let T : R n R m be a linear map Key observation: To describe T completely, it is enough to know the images T( e 1 ),, T( e n ) of the standard basis vectors e 1,, e n Reason:linearity The matrix A = [T( e 1 ) T( e n )] is called the matrix of the linear map T : R n R m (with respect to the standard basis) Verify that T( v) =Av for all v R n This way, each linear map T : R n R m can be described by a m n matrix Conversely, a m n matrix A determines a linear map L A : R n R m defined by L A ( v) =A v This sets up an one to one correspondence between the set of linear maps from R n R m and the set of m n matrices Verify that T( v) =A v for all v R 2