Math 5718 - Assignment 7
Spring 2002
Due April 23
You may work on assignments in groups and share ideas. You should also feel free to ask me questions about specific problems. If you reach an impasse, seek help so you can complete the solution. The goal of the assignments is to complete as many solutions as possible. However, it is imperative that you ultimately write up your own solutions and turn in only your own work. You may use symbolic algebra software, but please do not use linear algebra software. The calculations are reasonable.
  1. Please finish reading Chapter 6 of the text.
  2. Gram-Schmidt and related matters in ${\bf R}^3$. Consider the linearly independent vectors $v_1=(1,1,1)^T, v_2=(1,1,0)^T$, and $v_3=(0,1,0)^T$.
    1. Apply the Gram-Schmidt process to this set to produce a set of orthonormal vectors $\{e_1,e_2,e_3\}$.
    2. Express the vector $v=\sqrt{6}(1,-2,3)^T$ as a linear combination of $e_1,e_2,e_3$.
    3. Let $v_1, v_2$, and $v_3$ form the columns of a $3 \times 3$ matrix $A$. Use the result of part(a) to find $3 \times 3$ matrices $Q$ and $R$ such that $A=QR$.
    4. According to Theorem 6.45, given a linear functional $\phi$ on ${\bf R}^3$, it is possible to find a unique vector $u^\ast \in {\bf R}^3$ such that for all $v \in {\bf R}^3$, $\phi(v)=\langle v,u^\ast \rangle$. Find $u^\ast$ for the functional $\phi(v)=3v_1-2v_2+9v_3$.
  3. Gram-Schmidt and related matters in $P_2({\bf R})$. Consider the linearly independent set $v_1=1,v_2=x$, and $v_3=x^2$. The Legendre polynomials can be produced by applying the Gram-Schmidt process to this set with respect to the inner product $\langle u,v \rangle = \int_{-1}^1 uv\;dx$. Be sure to use symmetry arguments in evaluating the integrals that arise in the following problems.
    1. Find the first three Legendre polynomials $\{e_1,e_2,e_3\}$.
    2. Use orthogonality to express $v(x)=1-x$ as a linear combination of $e_1,e_2,e_3$.
    3. Consider the function $v(x)=e^{-x}$. Describe how you would find the best (least squares) approximation to $v$ in terms of $e_1,e_2,e_3$. Set up the integrals, but do not evaluate them.
    4. According to Theorem 6.45, given a linear functional $\phi$ on $P_2({\bf R})$, it is possible to find a unique vector (polynomial) $u^\ast \in P_2({\bf R})$ such that for all $v \in P_2({\bf R})$, $\phi(v)=\langle v,u^\ast \rangle$. Find $u^\ast$ for the functional $\phi(v)=v(\frac 12)$. Check your result with $v(x)=1-x$.
  4. Book problems. Please do problems 6, 27, 28, and 29 from Chapter 6 of the text.