Activity 6.3.4

Suppose that $L$ is the line in $\real^3$ defined by the vector $\wvec=\threevec{1}{2}{-2}$.

1. Find an orthonormal basis $\uvec$ for $L$.

2. Construct the matrix $Q = \begin{bmatrix}\uvec\end{bmatrix}$ and use it to construct the matrix $P$ that projects vectors orthogonally onto $L$.

3. Use your matrix to find $\bhat$, the orthogonal projection of $\bvec=\threevec111$ onto $L$.

4. Find $\rank(P)$ and explain its geometric significance.

The vectors $$\wvec_1 = \fourvec1111,\hspace{24pt} \wvec_2 = \fourvec011{-2}$$ form an orthogonal basis of $W$, a two-dimensional subspace of $\real^4$.

1. Use the projection formula from Projection formula to find $\bhat$, the orthogonal projection of $\bvec=\fourvec92{-2}3$ onto $W$.

2. Find an orthonormal basis $\uvec_1$ and $\uvec_2$ for $W$ and use it to construct the matrix $P$ that projects vectors orthogonally onto $W$. Check that $P\bvec = \bhat$, the orthogonal projection you found in the previous part of this activity.

3. Find $\rank(P)$ and explain its geometric significance.

4. Find a basis for $W^\perp$.

5. Find a vector $\bvec^\perp$ in $W^\perp$ such that $$\bvec = \bhat + \bvec^\perp.$$

6. If $Q$ is the matrix whose columns are $\uvec_1$ and $\uvec_2$, find the product $Q^TQ$ and explain your result.