Sheet1 Page 1 1.3 # 12ad For something like (a, b+c-2a, c-b,c) write it as a(1,-2,0,0) + b(0,1,-1,0) + c(0,1,1,1) and recognize the set as all linear combinations of 3 vectors [next week we will see that ANY span of a collection of vectors forms a subspace, but here you are supposed to verify that the sum and scalar multiple of such sums above is again such a sum] # 17 The expression for the general solution is again of the form in 12ad. # 19 Choose nice easy values for r,s (and who could be nicer than 0 or 1?), do this 4 times, and then check which parameters r,s give rise to these 4. 7.1 # 9 We said that each matrix has a UNIQUE RREf, but we said nothing about REF. Think about a matrix in lovely REF form, and think of a possible way to change it by an elementary row operation WHICH STILL KEEPS IT IN REF. Remember that REF has to have each nonzero row start with a leading 1 and these in stairstep order, so you probably can't scale or switch. #10 Look at the table of steps on page 357
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