100%(2)2 out of 2 people found this document helpful
This preview shows page 16 - 20 out of 185 pages.
Which linear combinations ofbracketleftbigg21bracketrightbiggandbracketleftbigg-11bracketrightbiggproducebracketleftbigg15bracketrightbigg?This example has the unique solutionx=2,y=3.•(2,3)is the (only) intersection of the two lines2x-y=1andx+y=5.•2bracketleftbigg21bracketrightbigg+3bracketleftbigg-11bracketrightbiggis the (only) linear combination producingbracketleftbigg15bracketrightbigg.Armin Straub[email protected]6
Pre-lecture:the shocking state of our ignoranceQ:How fast can we solveNlinear equations inNunknowns?Estimated cost of Gaussian elimination:squaresolid∗∗∗0∗∗∗0∗∗∗•to create the zeros below the pivot:on the order ofN2operations•if there isNpivots:on the order ofN·N2=N3op’s•A more careful count places the cost at∼13N3op’s.•For largeN, it is only theN3that matters.It says that ifN→10Nthen we have to work1000times as hard.That’s not optimal!We can do better than Gaussian elimination:•Strassen algorithm (1969):Nlog27=N2.807•Coppersmith–Winograd algorithm (1990):N2.375•Stothers–Williams–Le Gall (2014):N2.373IsN2possible? We have no idea!(better is impossible; why?)Good news for applications:(will see an example soon)•Matrices typically have lots of structure and zeroswhich makes solving so much faster.Armin Straub[email protected]1
Organizational•Help sessions in 441 AH: MW 4-6pm, TR 5-7pmReview•A system such as2x−y= 1x+y= 5can be written invectorform asxbracketleftbigg21bracketrightbigg+ybracketleftbigg−11bracketrightbigg=bracketleftbigg15bracketrightbigg.•The left-hand side is alinear combinationof the vectorsbracketleftbigg21bracketrightbiggandbracketleftbigg-11bracketrightbigg.The row and column pictureExample 1.We can think of the linear system2x−y= 1x+y= 5in two different geometric ways. Here, there is a unique solution:x=2,y=3.Row picture.•Each equation defines a line inR2.•Which points lie on the intersectionof these lines?•(2,3)is the (only) intersection ofthe two lines2x−y=1andx+y=5.1234512345Armin Straub[email protected]2
Column picture.•The system can be written asxbracketleftbigg21bracketrightbigg+ybracketleftbigg-11bracketrightbigg=bracketleftbigg15bracketrightbigg.•Which linear combinations ofbracketleftbigg21bracketrightbiggandbracketleftbigg-11bracketrightbiggproducebracketleftbigg15bracketrightbigg?•(2,3)are the coefficients of the(only) such linear combination.Minus3Minus2Minus10123412345Example 2.Consider the vectorsa1=103,a2=4214,a3=3610,b=−18−5.