Unformatted text preview: Example: Sensitivity of Linear Systems
Consider the vectors a1 = 0.999 1 , and a2 = 1 1.001 . Let’s let the matrix A have columns a1 and a2 : A= a1 , a 2 = 0.999 1 1 1.001 . You can check that A is nonsingular. Then a1 and a2 are linearly independent, and so any b ∈ R2 can be written in exactly one way as a linear combination b = x1 a1 + x2 a2 . The coeﬃcients x1 and x2 of this combination are the coordinates of the solution x = (x1 , x2 )t , of the matrix equation Ax = b. Now let’s take b to be b= 1.9989 2.0010 . When we solve Ax = b, we ﬁnd that (no rounding errors here) x= that is, b = 101.1 a1 − 99 a2 , Now suppose we round b to the nearest thousandths place: ˜= b 1.999 2.001 . 101.1 −99 . This small change in b can be measured: b − ˜ ∞ = 0.0001. Now how much of a1 and b ˜? Well, we solve Ax = ˜ to get a2 do we need to make b b x= that is, ˜ = 1 a1 + 1 a2 . b A change in b of about 10−4 gives a change in x of over 102 . This example was designed to make a point, you can see how it works by interpreting Ax = b as the intersection of 2 lines. Go ahead and plot the lines... 1 1 , ...
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 Fall '10
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 Linear Algebra, Vector Space, ax, matrix equation Ax

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