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The final exam for math 205a, focusing on finding the coefficients of a parabola that fits given data points and orthogonal projections. Students are required to solve systems of equations, find the inverse of a matrix, and determine the column space and null space of a matrix.
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1a) The three points (1, 5), (2, 3), and (3, −5) all lie on one parabola of the form y = a + bx + cx^2. What system of equations in the unknowns a, b and c needs to be solved in order to find a, b, and c?
1b) Write the above system in augmented matrix form, and proceed to put that matrix in row- reduced echelon form.
1c) What are the values of a, b, and c? So, what is the parabola?
1
2a) Refer to problem 1. Now set up the design matrix X, observation vector y and unknown- parameter vector β which corresponds to a best-fit parabola of the form y = β 1 x + β 2 x^2 (ie, a parabola with no constant term) for those three data points (1, 5), (2, 3), and (3, −5).
2b) Find β 1 and β 2. Show all steps, including any multiplications involving matrix transposes. Note: at the end of the work, you may find it easier to use the formula for the inverse of a 2× 2 matrix, rather than row reduction (to avoid working with fractions, keep the “1/det” outside; don’t multiply through by it.).
2c) In terms of the sum of residuals’ squares, how “far” is this best-fit parabola from the data? Show your work.
3e) Explain why finding a basis for Col(A))⊥^ is the same as finding a basis for Null(Col(AT^ )). Hint: Col(A))⊥^ consists of all vectors v which are ⊥^ to all the column vectors of A. That means c · v=0 for each column c of A. Columns of A turn into rows of AT^. When you find the product AT^ v, you multiply the rows of AT^ by the column v — that is, you’re finding dot products of what with what?
3f) Find a basis for Col(A))⊥.
. Explain why the columns of^ A^ form an orthogonal set.
4b) Find the projection of b =
onto the column space of^ A. Use the appropriate dot product
formula. Show all your work.
matrices of the form v =
a 0 3 a + 2b b
. Prove H is a subspace of M 2 × 2 or show by counterexample
that it is not.
7a) Find the characteristic polynomial of A.
7b) What are the eigenvalues of A and their multiplicities?
7c) Find bases for each of the two eigenspaces. Indicate which basis goes with which eigenvalue.
7d) Use the answers above to diagonalize A, that is, to write A = P DP −^1 where D is a diagonal matrix.
a b 0 b b 1
(^) Use simultaneuos row-reduction on A and I 3 to find A−^1. Under what
condition(s) will A−^1 not exist?
