Activity 6.4.2

Suppose that $W$ is a three-dimensional subspace of $\real^4$ with basis: $$\vvec_1 = \fourvec1111,\hspace{24pt} \vvec_2 = \fourvec1322,\hspace{24pt} \vvec_3 = \fourvec1{-3}{-3}{-3}\text{.}$$ We can see that this basis is not orthogonal by noting that $\vvec_1\cdot\vvec_2 = 8$. Our goal is to create an orthogonal basis $\wvec_1$, $\wvec_2$, and $\wvec_3$ for $W$.

To begin, we declare that $\wvec_1=\vvec_1$, and we call $W_1$ the line defined by $\wvec_1$.

1. Find the vector $\vhat_2$ that is the orthogonal projection of $\vvec_2$ onto $W_1$, the line defined by $\wvec_1$.

2. Form the vector $\wvec_2 = \vvec_2-\vhat_2$ and verify that it is orthogonal to $\wvec_1$.

3. Explain why $\laspan{\wvec_1,\wvec_2} = \laspan{\vvec_1,\vvec_2}$ by showing that any linear combination of $\vvec_1$ and $\vvec_2$ can be written as a linear combination of $\wvec_1$ and $\wvec_2$ and vice versa.

4. The vectors $\wvec_1$ and $\wvec_2$ are an orthogonal basis for a two-dimensional subspace $W_2$ of $\real^4$. Find the vector $\vhat_3$ that is the orthogonal projection of $\vvec_3$ onto $W_2$.

5. Verify that $\wvec_3 = \vvec_3-\vhat_3$ is orthogonal to both $\wvec_1$ and $\wvec_2$.

6. Explain why $\wvec_1$, $\wvec_2$, and $\wvec_3$ form an orthogonal basis for $W$.

7. Now find an orthonormal basis for $W$.