Linear Algebra with Applications, 10th Edition

Section 5.6 Exercises

1. For each of the following, use the Gram–Schmidt process to find an orthonormal basis for R(A):

   1. $A=\left[\begin{array}{cc}\hfill -1& \hfill 3\\ \hfill 1& \hfill 5\end{array}\right]$

   2. $A=\left[\begin{array}{cc}\hfill 2& \hfill 5\\ \hfill 1& \hfill 10\end{array}\right]$

2. Factor each of the matrices in Exercise 1 into a product QR, where Q is an orthogonal matrix and R is upper triangular.

3. Given the basis $\{{(1,2,-2)}^T,{(4,3,2)}^T,{(1,2,1)}^T\}$ for ${ℝ}^3$, use the Gram–Schmidt process to obtain an orthonormal basis.

4. Consider the vector space $C\left[-1,1\right]$ with the inner product defined by

   $$\langle f,g\rangle ={\int }_{-1}^{1}f(x)g(x)dx$$

   Find an orthonormal basis for the subspace spanned by 1, x, and $x^2$.

5. Let

   $$\begin{array}{ccc}A=\left[\begin{array}{cc}2& 1\\ 1& 1\\ 2& 1\end{array}\right]& \text{and}& \mathbf{b}=\left[\begin{array}{r}12\\ 6\\ 18\end{array}\right]\end{array}$$

   1. Use the Gram–Schmidt process to find an orthonormal basis for the column space of A.

   2. Factor A into a product QR, where Q has an orthonormal set of column vectors and R is upper ...