Pre-lecture: the shocking state of our ignorance

Transcription

1 Pre-lecture: the shocking state of our ignorance Q: How fast can we solve N linear equations in N unknowns? Estimated cost of Gaussian elimination: 0 0 to create the zeros below the pivot: on the order of N operations if there is N pivots: on the order of N N =N op s A more careful count places the cost at N op s For large N, it is only the N that matters It says that if N 0N then we have to work 000 times as hard That s not optimal! W e can do better than Gaussian elim ination: Strassen algorithm (969): N log7 =N 807 Coppersmith Winograd algorithm (990): N 7 Stothers Williams Le Gall (0): N 7 Is N possible? We have no idea! (better is im p ossible; why?) Good news for applications: Matrices typically have lots of structure and zeros (will see an exam ple soon) which makes solving so much faster

2 Organizational Help sessions in AH: MW -6pm, TR -7pm Review A system such as x y = x+y = can be written in vector form as x +y = The left-hand side is a linear combination of the vectors and The row and column picture Example We can think of the linear system x y = x+y = in two different geometric ways Here, there is a unique solution: x=, y= Row picture Each equation defines a line in R Which points lie on the intersection of these lines? (, ) is the (only) intersection of the two lines x y = and x + y=

3 Column picture The system can be written as x +y = Which linear combinations of and produce? (, ) are the coefficients of the (only) such linear combination 0 Example Consider the vectors a = 0, a =, a = 6 0, b= 8 Determine if b is a linear combination of a,a,a Solution Vector b is a linear combination of a,a,a if we can find weights x,x, x such that: x 0 +x +x 6 = 8 0 This vector equation corresponds to the linear system: x +x +x = +x +6x = 8 x +x +0x = Corresponding augmented matrix: Note that we are looking for a linear combination of the first three columns which

4 produces the last column Such a combination exists the system is consistent Row reduction to echelon form: Since this system is consistent, b is a linear combination of a,a,a It is consistent, because there is no row of the form b with b 0 Example In the previous example, express b as a linear combination of a,a,a Solution The reduced echelon form is: We read off the solution x =, x =, x =, which yields = Summary A vector equation x a +x a + +x m a m =b has the same solution set as the linear system with augmented matrix a a a m b In particular, b can be generated by a linear combination of a,a,,a m if and only if this linear system is consistent

5 The span of a set of vectors Definition The span of vectors v,v,,v m is the set of all their linear combinations We denote it by span{v,v,,v m } In other words, span{v,v,,v m } is the set of all vectors of the form where c,c,,c m are scalars c v +c v + +c m v m, Example (a) Describe span{ } geometrically The span consists of all vectors of the form α As points in R, this is a line (b) Describe span{, The span is all of R, a plane } geometrically That s because any vector in R can be written as x +x Let s show this without relying on our geometric intuition: let 0 0 b b b b b 0 b b is consistent any vector Hence, b b (c) Describe span{ Note that is a linear combination of, = and } geometrically Hence, the span is as in (a) Again, we can also see this after row reduction: let b b b 0 0 b b b b any vector is not consistent for all b b b b is in the span of and So the span consists of vectors b b only if b b =0 (ie b = b ) =b

6 A single (nonzero) vector always spans a line, two vectors v,v usually span a plane but it could also be just a line (if v =αv ) We will come back to this when we discuss dimension and linear independence Example 6 Is span {, } a line or a plane? Solution The span is a plane unless, for some α, =α Looking at the first entry, α =, but that does not work for the third entry Hence, there is no such α The span is a plane Example 7 Consider A=, 0 b= 8 7 Is b in the plane spanned by the columns of A? Solution b in the plane spanned by the columns of A if and only if is consistent To find out, we row reduce to an echelon form: From the last row, we see that the system is inconsistent Hence, b is not in the plane spanned by the columns of A 6

7 Conclusion and summary The span of vectors a,a,,a m is the set of all their linear combinations Some vector b is in span{a, a,, a m } if and only if there is a solution to the linear system with augmented matrix a a a m b Each solution corresponds to the weights in a linear combination of thea,a,, a m which gives b This gives a second geometric way to think of linear systems! 7