applied linear algebra

167 Homework 3

Due Monday November 13 in class.

Be sure to write your name and student ID clearly on your submission so we can give you credit for your work!

It is OK to work in teams on this homework but you must list who you worked with so the TA does not think you are copying somebody else’s work. It is OK to use any reference (Google, Wikipedia, Wolfram Alpha, books, research papers, your uncle’s niece etc...) to solve these problems, but you MUST properly cite your sources!

Question 1: Captain Conundrum falls off the leaning tower of Pisa and makes three (rather shaky) measurements of his velocity at three different times:

t (s) v (m/s) 1 11 2 19 3 31

He posits two possible theoretical models to explain his data

v = at + b and v = a(t−10/3) + b,

where a and b are coefficients he plans to find by fitting to data. Use the least squares method to find (a,b) for both models. In each case you will try to ap- proximate a linear system MX = V by MTMX = MTV and then invert MTM. Calculate the determinant of MTM for both cases. Finally compare and plot the results you find for these least square computations.

Question 2: Suppose that the only solution to MX = 0 is the zero vector. Explain why the matrix MTM is invertible.

Now let ε > 0. Compute the kernel of the matrix

M =

 1 1ε 0 0 ε

  .

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Check that the matrix MTM is invertible (hint: compute its determinant). Suppose that ε = 10−20. What will a computer using floating point arithmetic find for the determinant of MTM?

Question 3: Let (V,〈·, ·〉) be an inner product space with basis {v1,v2,v3,v4,v5)}. Explain how to use the Gram–Schmidt process to generate an orthonormal basis for V from this data (hint: you may need to look this up in your old linear algebra notes). Now pick any five linearly independent vectors in R5 that are not orthonor- mal with respect to the standard inner product given by the dot product. Apply the general result you just described to these to obtain an orthonormal basis for R5.

Question 4: Consider the rational numbers (Q, ||·||) as a normed space (a normed space is just a vector space with a norm) where || · || is the 2-adic norm described in class. Calculate

∣∣∣∣− 1 3

∣∣∣∣. (Hint, try writing −1 3

as a formal power series in the “small” parameter 2.)

Question 5: Let (V, || · ||) be a normed space where V is a finite dimensional vector space. Now suppose

L : V linear−→ V .

The spectral radius of L is the absolute value of its largest eigenvalue. Show that the spectral radius of L is always less than or equal to the operator norm ||L|| induced by || · ||.

Question 6: A matrix is said to be banded with bandwidth w if it only has non- zero entries along a band of width w around its diagonal. For example, the banded (“tridiagonal”) matrix 

 1 1 0 0 0 0 1 2 2 0 0 0 0 2 3 3 0 0 0 0 3 4 4 0 0 0 0 4 5 5 0 0 0 0 5 6

 

has bandwith 3. Let M = (mij) be an n×n square matrix whose only non-zero elements are

mii = i , mi,i+1 = i , mi+1,i = i .

Is this matrix banded and if so what is its bandwidth? Compute its determinant.

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(Hint: use matlab to figure out the answer for a few low values of n and establish a pattern, then show why this pattern holds in general.)

Question 7: Many problems in calculus can be handled numerically by replacing the real number line with a finite set of points S = {1,2,3, . . . ,N}, and the derivative operator with a difference operator. Let V = RS and define the operator

d : V linear−→ V

by df(i) = f(i)−f(i−1) if i > 1 and df(1) = f(1) .

Write the matrix of d in the basis {e1, . . . ,eN} where ej(i) = δij (here δij is the Kronecker delta symbol). Is d banded, and if so what is its bandwidth? Use matrix multiplication to compute the matrix of d2. Is this matrix also banded and if so what is its bandwidth? What is the bandwidth of dk for any integer k? Try to compute the inverse of d (again some matlab exploration is a good idea). What is the formula for d−1f(i)? How do you interpret this result in terms of standard calculus?

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