Math 641-600 — Fall 2015

Assignments

Assignment 1 - Due Wednesday, September 9, 2015.

• Do the following problems.
1. Section 1.1: 3(c), 5, 7(a), 8
2. Section 1.4: 3
3. Let $U$ be a subspace of an inner product space $V$, with the inner product and norm being $\langle\cdot,\cdot \rangle$ and $\|\cdot\|$. Also, let $v$ be in $V$. (Do not assume that $U$ is finite dimensional or use arguments requiring a basis.)
1. Fix $v\in V$. Show that if there exists $p\in U$ such that $p$ satisfies either (i) $\min_{u\in U}\|v-u\| = \|v-p\|$ or (ii) $v-p\in U^\perp$, then it satisfies both (i) and (ii). Moreover, if $p$ exists, it is unique.
2. Suppose $p$ exists for every $v\in V$. Since $p$ is uniquely determined by $v$, we may define a map $P: V \to U$ via $Pv:=p$. Show that $P$ is a linear map and that $P$ satisfies $P^2 = P$. ($P$ is called an orthogonal projection. The vector $p$ is the orthogonal projection of $v$ onto $U$.)
3. If the projection $P$ exists, show that for all $w,z\in V$, $\langle Pw,z\rangle = \langle Pw,Pz\rangle= \langle w,Pz\rangle$. Use this to show that $U^\perp= \{w\in V\colon Pw=0\}$.
4. Suppose that the projection $P$ exists. Show that $V=U\oplus U^\perp$, where $\oplus$ indicates the direct sum of the two spaces.

4. Let $U$ and $V$ be as in the previous exercise. Suppose that $U$ is finite dimensional and that $B=\{u_1,u_2,\ldots,u_n\}$ is an ordered basis for $U$. In addition, let $G$ be the $n\times n$ matrix with entries $G_{jk}= \langle u_k,u_j\rangle$.
1. Let $v\in V$ and $d_k := \langle v,u_k\rangle$. Show that $p$ exists for every $v$ and is given by $p=\sum_j x_j u_j\in U$, where the $x_j$'s satisfy the normal equations, $d_k = \sum_j G_{kj}x_j$. (Hint: use exercise 3 above.)
2. Explain why the projection $P$ exists. Show that if B is orthonormal, then $Pv=\sum_j \langle v,u_j\rangle u_j$.

Assignment 2 - Due Wednesday, September 16, 2015.

• Read sections 2.1 and 2.2
• Do the following problems.
1. Section 1.2: 9, 10(a,b)
2. Section 1.3: 2(b)
3. Section 1.4: 4
4. Let V be an n dimensional vector space and suppose L:V→V is linear.
1. Let E={φ1,.., φ1} be a basis for V and let A be the matrix of L relative to E. Show that the characteristic polynomial pA(λ) := det(A − λ I) is independent of the choice of E, and so pL(λ) := pA(λ) is well defined, in the sense that it is independent of the choice of basis for V.
2. Use the previous part to show that Trace(L) := Trace(A) and det(L) := det(A) are also well defined,
3. Let $L:P_2\to P_2$ be given by $L(p)= \big((1-x^2)p'\big)' + 7p$. Find Trace(L) and det(L).
5. Let A and B be n×n matrices. Suppose that the range of B, Range(B), is an invariant subspace for A. Show that there is an n×n matrix X such that AB = BX.

6. (This is a generalization of Keener's problem 1.3.5.) Let $A$ be a self-adjoint matrix with eigenvalues $\lambda_1\ge \lambda_2,\ldots,\ge \lambda_n$. Show that for $2\le k < n$ we have $\max_U \sum_{j=1}^k \langle Au_j,u_j \rangle =\sum_{j=1}^k \lambda_j,$ where $U=\{u_1,\ldots,u_k\}$ is any o.n. set. (Hint: Put $A$ in diagonal form and use a judicious choice of $B$.)

7. Let U be a unitary, n×n matrix. Show that the following hold.
1. < Ux, Uy > = < x, y >
2. The eigenvalues of U all lie on the unit circle, |λ|=1.
3. Show that U is diagonalizable. (Hint: follow the proof for the self-adjoint case.)

8. Let $k(x,y) = x+ 3x^2y + xy^2$ and $\langle f,g\rangle=\int_{-1}^1 f(x)g(x)(1+x^2)dx$. Consider the operator $Lu=\int_{-1}^1 k(x,y) u(y)dy$. In the notes, we have shown that $L:P_2\to P_2$.
1. Relative to the inner product above, find $L^\ast$ and $\text{Null}(L^\ast)$.
2. Find a condition on $q\in P_2$ for which $Lp=q$ always has a solution. Is this different from what was in the notes?

Assignment 3 - Due Wednesday, September 23, 2015.

Assignment 4 - Due Friday, October 2, 2015.

Assignment 5 - Due Friday, October 9.

Assignment 6 - Due Friday, October 30.

Assignment 7 - Due Friday, November 6.

• Read sections 3.1 and 3.2.
• Do the following problems.

1. Section 2.2: 25(a,b), 26(b), 27(a)

2. Let $H_0$ be the set of all $f\in C^{(0)}[0,1]$ such that $f(0)=f(1)=0$ and that $f'$ is piecewise continuous. Show that $\langle f,g\rangle_{H_0} :=\int_0^1f'(x)g'(x)dx$ defines a real inner product on $H_0$.

3. We want to use the Galerkin method to numerically solve the boundary value problem (BVP):  −u" = f(x), u(0) = u(1) = 0, f ∈ C[0,1]

1. Weak form of the problem. Let H0 be as in the previous problem. Suppose that $v\in H_0$. Multiply both sides of the eqaution above and use integration by parts to show that $\langle u,v\rangle_{H_0} = \langle f,v\rangle_{L^2[0,1]}$. This is called the weak'' form of the BVP.

2. Conversely, suppose that u ∈ H0 is also in C(2)[0,1] and that u satisfies
⟨u,v⟩H0 = ∫01 f(x) v(x) dx for all v ∈ H0.
Show that u satisfies the BVP.

3. Consider S0 := {s ∈S1/n(1,0) : s(0)=s(1)=0}. Show that S0 is spanned by φj(x) := N2(nx-j+1), j = 1 ... n-1. (Here, N2(x) is the linear B-spline.)

4. Show that the least-squares approximation s ∈ S0 to the solution u is given by s = ∑j αjφj(x), where the αj's satisfy Gα = β, with
βj = ⟨ y,φjH0 = ∫01 f(x) φj(x) dx, j=1 ... n-1 and Gkj = ⟨ φj, φkH0.

5. Show that Gkj = ⟨ φj, φkH0 is given by
Gj,j = 2n, j = 1 ... n-1
Gj,j-1 = - n, j = 2 ... n-1
Gj,j+1 = - n, j = 1 ... n-2
Gj,k = 0, all other possible k.

Assignment 8 - Due Friday, November 13.

Assignment 9 - Due Friday, November 20.

• Read sections 3.5 and 3.6.
• Do the following problems.

1. Section 3.4: 2(b)

2. Consider the real Hilbert space H0 := {v ∈ H1[0,1] : v(0) = v(1) = 0}, with the inner product < f,g >H0 = ∫01f ′(x) g ′(x)dx. Show that there is a function gy in H0 for which f(y) = < f,gy >H0. (gy is called a reproducing kernel and H0 is a reproducing kernel Hilbert space.) Hint: Use problem 4, HW3 to show that the Φy(f) := f(y) is a bounded linear functional on H0.

3. Let $L$ be a bounded operator on a Hilbert space $H$. Show that the closure of the the range of $L$ satisfies $\overline{R(L)} = N(L^\ast)^\perp$. (Hint: Follow the proof of the Fredholm alternative, which is just the special case where $R(L)$ is closed.)

4. A sequence {fn} in a Hilbert space H is said to be weakly convergent to f ∈ H if and only if lim n → ∞ < fn,g> = < f,g> for every g∈H. When this happens, we write f = w-lim fn. For example, if {φn} is any orthonormal sequence, then φn converges weakly to 0. You are given that every weakly convergent sequence is a bounded sequence (i.e. there is a constant C such that ||fn|| ≤ C for all n). Prove the following:
Let K be a compact linear operator on a Hilbert space H. If fn converges weakly to f, then Kfn converges to Kf — that is, lim n → ∞ || Kfn - Kf || = 0.
Hint: Suppose this doesn't happen, then there will be a subsequence of {fn}, say {fnk}, such that || Kfnk - Kf || ≥ ε for all k. Use this and the compactness of K to arrive at a contradiction. We remark that the converse is also true. If a bounded linear operator $K$ maps weakly convergent sequences into convergent sequences, then $K$ is compact.

5. Consider the Hilbert space $\ell^{\,2}$. Let $S=\{\{a_j\}_{j=1}^\infty \in \ell^{\,2}\colon \sum_{j=1}^\infty (1+j^2)\,|a_j|^2 < 1 \}$. Show that $S$ is a precompact subset of $\ell^{\,2}$.
6. Let L be a bounded self-adjoint linear operator on a Hilbert space $\mathcal H$. Show that these two formulas for $\|L\|$ are equivalent:
1. $\|L\| = \sup \{\|Lu\| : u \in {\mathcal H},\ \|u\| = 1\}$
2. $\|L\| = \sup \{|\langle Lu,u\rangle| : u\in {\mathcal H},\ \|u\|=1\}$

Assignment 10 - Due Monday, November 30.

Extra Problems - These are not to be handed in.

1. Let $L$ be in $\mathcal B (\mathcal H)$.
1. Show that $\|L^k\| \le \|L\|^k$, $k=2,3,\ldots$.
2. (We did this in class.) Let $|\lambda| \|L\|<1$. Show that $\big\|(I - \lambda L)^{-1} - \sum_{k=0}^{n-1}\lambda^k L^k\big\| \le \frac{|\lambda|^k \|L\|^k}{1 - |\lambda| \|L\|}.$
3. Let $L$ be as in problem 6, HW8. Estimate how many terms it would require to approximate $(I - \lambda L)^{-1}$ to within $10^{-8}$, if $|\lambda|\le 0.1$.
2. Use Newton's method (see text, problem 3.6.3) to approximate the cube root of 2. Show that the method converges.
3. Section 4.1: 6
4. Section 4.2: 1, 4, 8
5. Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
1. Show that the Green's function for this problem is $G(x,y)=\left\{ \begin{array}{rl} -(2y-1)x, & 0 \le x < y \le 1\\ -(2x-1)y, & 0 \le y< x \le 1. \end{array} \right.$
2. Verify that $0$ is not an eigenvalue for $Kf(x) := \int_0^1G(x,y)f(y)dy$.
3. Show the orthonormal set of eigenfunctions for $L$ form a complete set in $L^2[0,1]$. (Hint: use tthe results from problem 4, HW10.

Updated 12/8/2015.