Activity 6.4.4

Suppose that $A$ is the $4\times3$ matrix whose columns are $$\vvec_1 = \fourvec1111,\hspace{24pt} \vvec_2 = \fourvec1322,\hspace{24pt} \vvec_3 = \fourvec1{-3}{-3}{-3}\text{.}$$ These vectors form a basis for $W$, the subspace of $\real^4$ that we encountered in this activity. Since these vectors are the columns of $A$, we have $\col(A) = W$.

1. When we implemented Gram-Schmidt, we first found an orthogonal basis $\wvec_1$, $\wvec_2$, and $\wvec_3$ using $$\begin{aligned} \wvec_1 & = \vvec_1 \\ \wvec_2 & = \vvec_2 - \frac{\vvec_2\cdot\wvec_1}{\wvec_1\cdot\wvec_1}\wvec_1 \\ \wvec_3 & = \vvec_3 - \frac{\vvec_3\cdot\wvec_1}{\wvec_1\cdot\wvec_1}\wvec_1 - \frac{\vvec_3\cdot\wvec_2}{\wvec_2\cdot\wvec_2}\wvec_2\text{.} \\ \end{aligned}$$ Use these expressions to write $\vvec_1$, $\vvec_2$, and $\vvec_3$ as linear combinations of $\wvec_1$, $\wvec_2$, and $\wvec_3$.

2. We next normalized the orthogonal basis $\wvec_1$, $\wvec_2$, and $\wvec_3$ to obtain an orthonormal basis $\uvec_1$, $\uvec_2$, and $\uvec_3$.

   Write the vectors $\wvec_i$ as scalar multiples of $\uvec_i$. Then use these expressions to write $\vvec_1$, $\vvec_2$, and $\vvec_3$ as linear combinations of $\uvec_1$, $\uvec_2$, and $\uvec_3$.

3. Suppose that $Q = \left[ \begin{array}{ccc} \uvec_1 & \uvec_2 & \uvec_3 \end{array} \right]$. Use the result of the previous part to find a vector $\rvec_1$ so that $Q\rvec_1 = \vvec_1$.

4. Then find vectors $\rvec_2$ and $\rvec_3$ such that $Q\rvec_2 = \vvec_2$ and $Q\rvec_3 = \vvec_3$.

5. Construct the matrix $R = \left[ \begin{array}{ccc} \rvec_1 & \rvec_2 & \rvec_3 \end{array} \right]$. Remembering that $A = \left[ \begin{array}{ccc} \vvec_1 & \vvec_2 & \vvec_3 \end{array} \right]$, explain why $A = QR$.

6. What is special about the shape of $R$?

7. Suppose that $A$ is a $10\times 6$ matrix whose columns are linearly independent. This means that the columns of $A$ form a basis for $W=\col(A)$, a 6-dimensional subspace of $\real^{10}$. Suppose that we apply Gram-Schmidt orthogonalization to create an orthonormal basis whose vectors form the columns of $Q$ and that we write $A=QR$. What are the shape of $Q$ and what the shape of $R$?