University:
Amherst College
Course:
MATH 272 | Linear Algebra with Applications
Academic year:

2022
Views:

471

Pages:

2
Author:

theresafu92247x

Math 272 Midterm 2 1. Short answer questions. No explanations are necessary. (a) Give a basis for each of the following vector spaces. • R3 • M2×2 (the space of 2 × 2 matrices) • P2 (the space of polynomials of degree at most 2) (b) Suppose that B and B 0 are two bases for a vector space V , and the change of basis matrix is 3 −4 B0 [I]B = . 4 3 1 Let ~v ∈ V be a vector such that [~v ]B = . Determine [~v ]B 0 . 1 (c) Suppose that T : R3 → R4 is a linear transformation, and that the nullspace N (T ) is 1-dimensional. What is the dimension of the range R(T )? 1 1 3 1 0 for R2 . Find the change of , and B = 2. Consider the two bases B = , −1 1 3 1 0 basis matrix [I]B B . 3. Consider the following three vectors in R4 .     1 2 2 4   ~u =  ~v =  0 1 2 4   3 6  w ~ = 1 6 Denote by W the span of {~u, ~v , w}. ~ (a) Find a basis of W . (b) What is the dimension of W? (c) Find an orthonormal basis for W . (Recall that a basis is called orthonormal if any two vectors in the basis are orthogonal, and each vector has norm equal to 1.)   1 1  (d) What element of W is closest to the vector ~b =  1? 1 (In other words, which element ~x of W minimizes k~x − ~bk?) 4. Suppose that V is an inner product space, and ~u, ~v are two nonzero vectors in V such that h~u, ~v i = 0. Prove, directly from the axioms of an inner product, that k~u + 2~v k > k~uk. 5. Suppose that T is the linear transformation represented by the following matrix.   1 2 1 0 0 0 1 1  A= 1 2 2 1 0 0 2 2 Math 272 Midterm 2 (a) Find a basis for the range R(T ). (b) Find a basis for the null space N (T ). 6. Suppose that f (x) is a continuous function on the interval [−π, π] that has been measured in a laboratory. You have reason to believe that the function should be approximately equal to some linear combination of sin(x) and cos(x). Using your experimental data, you compute the following integrals. Z π f (x) sin x dx = 7 Z −π π f (x) cos x dx = 13 −π You may also use, without proof, the values of the following integrals. Z π sin2 x dx = π Z π −π sin x cos x dx = 0 −π Z π cos2 x dx = π −π (a) Let V = C (0) [−π, π] denote the vector space of continuous functions on [−π, π]. Define an inner product on V , and identify each of the five integrals above as an inner product hf, gi of two elements f, g of V . (b) Suppose that we choose two constants c1 , c2 and define a function g(x) as follows: g(x) = c1 sin x + c2 cos x. Rπ Find the values of c1 , c2 that will ensure that −π (f (x) − g(x))2 dx is as small as possible.

Math 272 Midterm 2 Sample

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