Homework Chapter 6 UTK – M251 – Matrix Algebra Fall 2003, Jochen Denzler, MWF 1:25–2:15, Ayres 318

1. Which of the following are inner products on ℝ^{3}? For those that are not, determine which of the properties does/do not hold.

(a) ⟨u, v⟩ = 2u_{1}v_{1} + 3u_{2}v_{2}

(b) ⟨u, v⟩ = 2u_{1}v_{1} + 3u_{2}v_{2} + 3u_{3}v_{3} − 2u_{2}v_{3} − 2u_{3}v_{2}

(c) ⟨u, v⟩ = 2u_{1}v_{1} + 3u_{2}v_{2} − u_{3}v_{3}

(d) ⟨u, v⟩ = (u_{1}^{2} + u_{2}^{2} + u_{3}^{2})(v_{1}^{2} + v_{2}^{2} + v_{3}^{2})^{2}

2. If A is an invertible n × n matrix, then the formula ⟨u, v⟩ := (Au) · (Av) gives an inner product, which is called the inner product generated by A. Calculate the inner product

generated by [ \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ] of the vectors u = [ \begin{matrix}1 \\ 1 \end{matrix} ] and v = [ \begin{matrix}1 \\ −1 \end{matrix} ].

3. In this problem, let ∥w∥ denote not the euclidean norm, but the norm obtained from the inner product generated by the matrix A from the previous problem.

What condition do x and y, the components of the vector w have to satisfy such that ∥w∥ = 3 holds? Draw the curve ∥w∥ = 3 in the plane in a good, to-scale figure. If ∥w∥ were the euclidean norm, then the curve would be a circle of radius 3. What kind of curve did you come up with in the present case? Suggested plot window: x ∈ [−8, 8], y ∈ [−5, 5].

4. On the space ℝ^{3×3}, we take the inner product ⟨A, B⟩ := tr A^{T} B. Based on this inner

product, what is the norm of the matrix [ \begin{matrix}a & b & c \\ d & e & f \\ g & h & i \end{matrix} ]?

Before calculating too much, remember that in a certain matrix product you are about to compute, not all of the entries need to be computed.

5. To show that tr AB is not an inner product, even for square matrices, find an example of a 2 × 2 matrix A for which tr(A^{2}) < 0.

6. In the vector space C^{0}[0, 2π] of all continuous functions defined on the interval [0, 2π], we use the inner product ⟨f, g⟩ := ∫_{0}^{2π} f(x)g(x) dx. Write out explicitly what the Cauchy-Schwarz inequality says in the case f(x) = x and g(x) = sin x.

Also check this particular instance of the Cauchy-Schwarz inequality by explicit evaluation of all integrals. — I do expect Math, Phys and Engr majors to be able to do these integrals without help, but since this is not a calculus class, and for the benefit of those who have not used calculus for a while, I give some derivative-antiderivative formulas for your convenience:

\frac{d}{dx}(sin x − x cos x) = x sin x , \frac{d}{dx} \frac{1}{2}(x − sin x cos x) = sin^{2} x

7. Continuing the previous problem, what is the “angle” between the functions f and g, in the sense determined by the inner product given there. — Note: If you have even the slightest gut feeling that this “angle” has anything to do with any angle you might see when you graph the function f and g, be assured emphatically that this is not the case.

8. For m = 0, 1, 2, 3, …, we define the functions c_{m} ∈ C^{0}[0, 2π] by c_{m}(x) := cos mx. For m = 1, 2, 3, …, we define the functions s_{m} ∈ C^{0}[0, 2π] by s_{m}(x) := sin mx. We share this workload among the class. Everybody select at random one of the three (equally easy/difficult) parts as homework.

(a) Show that ⟨c_{m}, c_{n}⟩ = 0 if m ≠ n.

(b) Show that ⟨s_{m}, s_{n}⟩ = 0 if m ≠ n.

(c) Show that ⟨c_{m}, s_{n}⟩ = 0.

You may use the following trig’ formulas (which I would not expect you to have memorized):

(cos u)(cos v) = \frac{1}{2}(cos(u + v) + cos(u − v))
(sin u)(sin v) = \frac{1}{2}(cos(u − v) − cos(u + v))
(sin u)(cos v) = \frac{1}{2}(sin(u + v) + sin(u − v))

9. Refer to the previous problem, and have a look back at hwk 10 from Ch. 5. Show that the set S = {c_{0}, c_{1}, c_{2}, …, c_{100}, s_{1}, s_{2}, …, s_{100}} in C^{0}[0, 2π] is linearly independent. — What is the dimension of C^{0}[0, 2π]? (Possible answers are either a nonnegative integer, or “infinite-dimensional”.)

10. In ℝ^{4}, with the euclidean inner product (dot product), let U be the subspace spanned by the vectors [1, 1, 1, 1]^{T} and [−1, 0, 1, 2]^{T}. Find a basis for the orthogonal complement U^{⊥} of U.

11. Same question, but this time, take the inner product ⟨u, v⟩ := u_{1}v_{1}+2u_{2}v_{2}+3u_{3}v_{3}+u_{4}v_{4} instead of the dot product.

(I’d like to stress that the U^{⊥} in this problem refers to a different space than the U^{⊥} in the previous problem, because the meaning of the symbol ^{⊥} depends on the inner product chosen. This is why I give two so similar problems)

12. In ℝ^{3} with the dot product, use the Gram–Schmidt process to obtain an orthonormal basis from the basis S = {[2, 2, 1]^{T} , [1, 2, 2]^{T} , [2, −1, 2]^{T} }.

13. In the vector space P_{3} of polynomials of degree 3 or less, choose the inner product ⟨p, q⟩ := ∫_{−1}^{1} p(x)q(x) dx. Use the Gram–Schmidt process to obtain an orthogonal basis {p_{0}, p_{1}, p_{2}, p_{3}} from the standard basis S = {1, x, x^{2}, x^{3}}. Normalize the vectors (i.e., polynomials) such that ∥p_{0}∥^{2} = 2, ∥p_{1}∥^{2} = \frac{2}{3}, ∥p_{2}∥^{2} = \frac{2}{5}, ∥p_{3}∥^{2} = \frac{2}{7}.

(Note: This is a convenient normalization that avoids square roots in the coefficients.

— The polynomials you obtain are called Legendre polynomials, and the reason why they deserve to be given a name is exactly that they form an orthogonal basis. — You are not required to memorize their name, though. — Here is one piece of information about the polynomials p_{j} that I provide for the sole purpose that you can check your calculation in each step against possible calculational erors: p_{j}(1) = 1 for all j.)

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