# Which linear combinations of bracketleftbigg 2 1

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Which linear combinations of bracketleftbigg 2 1 bracketrightbigg and bracketleftbigg - 1 1 bracketrightbigg produce bracketleftbigg 1 5 bracketrightbigg ? This example has the unique solution x =2 , y =3 . (2 , 3) is the (only) intersection of the two lines 2 x - y =1 and x + y =5 . 2 bracketleftbigg 2 1 bracketrightbigg +3 bracketleftbigg - 1 1 bracketrightbigg is the (only) linear combination producing bracketleftbigg 1 5 bracketrightbigg . Armin Straub [email protected] 6
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: squaresolid 0 0 to create the zeros below the pivot: on the order of N 2 operations if there is N pivots: on the order of N · N 2 = N 3 op’s A more careful count places the cost at 1 3 N 3 op’s. For large N , it is only the N 3 that matters. It says that if N 10 N then we have to work 1000 times as hard. That’s not optimal! We can do better than Gaussian elimination: Strassen algorithm (1969): N log 2 7 = N 2.807 Coppersmith–Winograd algorithm (1990): N 2.375 Stothers–Williams–Le Gall (2014): N 2.373 Is N 2 possible? We have no idea! (better is impossible; why?) Good news for applications: (will see an example soon) Matrices typically have lots of structure and zeros which makes solving so much faster. Armin Straub [email protected] 1
Organizational Help sessions in 441 AH: MW 4-6pm, TR 5-7pm Review A system such as 2 x y = 1 x + y = 5 can be written in vector form as x bracketleftbigg 2 1 bracketrightbigg + y bracketleftbigg 1 1 bracketrightbigg = bracketleftbigg 1 5 bracketrightbigg . The left-hand side is a linear combination of the vectors bracketleftbigg 2 1 bracketrightbigg and bracketleftbigg - 1 1 bracketrightbigg . The row and column picture Example 1. We can think of the linear system 2 x y = 1 x + y = 5 in two different geometric ways. Here, there is a unique solution: x =2 , y =3 . Row picture. Each equation defines a line in R 2 . Which points lie on the intersection of these lines? (2 , 3) is the (only) intersection of the two lines 2 x y =1 and x + y =5 . 1 2 3 4 5 1 2 3 4 5 Armin Straub [email protected] 2
Column picture. The system can be written as x bracketleftbigg 2 1 bracketrightbigg + y bracketleftbigg - 1 1 bracketrightbigg = bracketleftbigg 1 5 bracketrightbigg . Which linear combinations of bracketleftbigg 2 1 bracketrightbigg and bracketleftbigg - 1 1 bracketrightbigg produce bracketleftbigg 1 5 bracketrightbigg ? (2 , 3) are the coefficients of the (only) such linear combination. Minus 3 Minus 2 Minus 1 0 1 2 3 4 1 2 3 4 5 Example 2. Consider the vectors a 1 = 1 0 3 , a 2 = 4 2 14 , a 3 = 3 6 10 , b = 1 8 5 .