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

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

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.

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

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

Let

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 triangular.

3. Solve the least squares problem $A\text{x}=\mathbf{b}$.

Repeat Exercise 5 using

Given ${\mathbf{x}}_1=\frac{1}{2}{(1,1,1,-1)}^T$ and ${\mathbf{x}}_2=\frac{1}{6}{(1,1,3,5)}^T$, verify that these vectors form an orthonormal set in ${ℝ}^4$. Extend this set to an orthonormal basis for ${ℝ}^4$ by finding an orthonormal basis for the null space of

[Hint: First find a basis for the null space and then use the Gram–Schmidt process.]

Use the Gram–Schmidt process to find an orthonormal basis for the subspace of ${ℝ}^4$ spanned by ${\mathbf{x}}_1={(4,2,2,1)}^T,{\mathbf{x}}_2={(2,0,0,2)}^T$, and ${\mathbf{x}}_3={(1,1,-1,1)}^T$.

Repeat Exercise 8 using the modified Gram–Schmidt process and compare answers.

Let A be an $m\times 2$ matrix. Show that if both the classical Gram–Schmidt process and the modified Gram–Schmidt process are applied to the column vectors of A, then both algorithms will produce the exact same QR factorization, even when the computations are carried out in finite-precision arithmetic (i.e., show that both algorithms will perform the exact same arithmetic computations).

Let A be an $m\times 3$ matrix. Let QR be the QR factorization obtained when the classical Gram–Schmidt process is applied to the column vectors of A, and let $\tilde{Q}\tilde{R}$ be the factorization obtained when the modified Gram–Schmidt process is used. Show that if all computations were carried out using exact arithmetic, then we would have

and show that when the computations are done in finite-precision arithmetic, ${\tilde{r}}_{23}$ will not necessarily be equal to $r_{23}$ and, consequently, ${\tilde{r}}_{33}$ and ${\tilde{\mathbf{q}}}_3$ will not necessarily be thesameas $r_{23}$ and ${\mathbf{q}}_3$.

What will happen if the Gram–Schmidt process is applied to a set of vectors $\{{\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\}$, where ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$ are linearly independent, but ${\mathbf{v}}_3\in \text{Span}({\mathbf{v}}_1,{\mathbf{v}}_2)$? Will the process fail? If so, how? Explain.

Let A be an $m\times n$ matrix of rank n and let $\mathbf{b}\in {ℝ}^m$. Show that if Q and R are the matrices derived from applying the Gram–Schmidt process to the column vectors of A and

is the projection of b onto R(A), then

1. $\mathbf{c}=Q^T\mathbf{b}$

2. $\mathbf{p}=Q{Q}^T\mathbf{b}$

3. $Q{Q}^T=A(A^{T}A{)}^{-1}{A}^T$

Let U be an m-dimensional subspace of ${ℝ}^n$ and let V be a k-dimensional subspace of U, where $0<k<m$.

1. Show that any orthonormal basis

   $$\{{\mathbf{v}}_1,{\mathbf{v}}_2,\mathrm{\dots },{\mathbf{v}}_k\}$$

   for V can be expanded to form an orthonormal basis $\{{\mathbf{v}}_1,{\mathbf{v}}_2,\mathrm{\dots },{\mathbf{v}}_k,{\mathbf{v}}_{k+1},\mathrm{\dots },{\mathbf{v}}_m\}$ for U.

2. Show that if $W=\text{Span}({\mathbf{v}}_{k+1},{\mathbf{v}}_{k+2},\mathrm{\dots },{\mathbf{v}}_m)$, then $U=V\oplus W$

(Dimension Theorem) Let U and V be subspaces of ${ℝ}^n$. In the case that $U\cap V=\{\mathbf{0}\}$, we have the following dimension relation:

(See Exercise 18 in Section 3.4.) Make use of the result from Exercise 14 to prove the more general theorem