Orthogonal bases and projections

1. 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.

2. The vectors form an orthogonal basis of $W$, a two-dimensional subspace of $\real^4$.

   1. Use the projection fo…

Consider the vectors

1. Explain why these vectors form an orthogonal basis for $\real^3$.

2. Suppose that $A=\begin{bmatrix} \wvec_1 & \wvec_2 & \wvec_3 \end{bmatrix}$ and evaluate the product $A^TA$. Why is this product a diagonal matrix and what is the significance of the diagonal entries?

3. Express the vector $\bvec=\threevec{-3}{-6}3$ as a linear combination of $\wvec_1$, $\wvec_2$, and $\wvec_3$.

4. Multiply the vectors $\wvec_1$, $\wvec_2$, $\wvec_3$ by appropriate scalars to find an orthonormal basis $\uvec_1$, $\uvec_2$, $\uvec_3$ of $\real^3$.

5. If $Q=\begin{bmatrix} \uvec_1 & \uvec_2 & \uvec_3 \end{bmatrix}$, find the matrix product $QQ^T$ an…

Suppose that form an orthogonal basis for a subspace $W$ of $\real^4$.

1. Find $\bhat$, the orthogonal projection of $\bvec=\fourvec{2}{-1}{-6}{7}$ onto $W$.

2. Find the vector $\bvec^\perp$ in $W^\perp$ such that $\bvec = \bhat + \bvec^\perp$.

3. Find a basis for $W^\perp$. and express $\bvec^\perp$ as a linear combination of the basis vectors.

Consider the vectors

1. If $L$ is the line defined by the vector $\wvec_1$, find the vector in $L$ closest to $\bvec$. Call this vector $\bhat_1$.

2. If $W$ is the subspace spanned by $\wvec_1$ and $\wvec_2$, find the vector in $W$ closest to $\bvec$. Call this vector $\bhat_2$.

3. Determine whether $\bhat_1$ or $\bhat_2$ is closer to $\bvec$ and explain why.

Suppose that form the basis for a plane $W$ in $\real^3$.

1. Find a basis for the line that is the orthogonal complement $W^\perp$.

2. Given the vector $\bvec=\threevec6{-6}2$, find $\yvec$, the orthogonal projection of $\bvec$ onto the line $W^\perp$.

3. Explain why the vector $\zvec = \bvec-\yvec$ must be in $W$ and write $\zvec$ as a linear combination of $\vvec_1$ and $\vvec_2$.

Determine whether the following statements are true or false and explain your thinking.

1. If the columns of $Q$ form an orthonormal basis for a subspace $W$ and $\wvec$ is a vector in $W$, then $QQ^T\wvec = \wvec$.

2. An orthogonal set of vectors in $\real^8$ can have no more than 8 vectors.

3. If $Q$ is a $7\times5$ matrix whose columns are orthonormal, then $QQ^T = I_7$.

4. If $Q$ is a $7\times5$ matrix whose columns are orthonormal, then $Q^TQ = I_5$.

5. If the orthogonal projection of $\bvec$ onto…

Suppose that $Q$ is an orthogonal matrix.

1. Remembering that $\vvec\cdot\wvec=\vvec^T\wvec$, explain why

2. Explain why $\len{Q\xvec} = \len{\xvec}$.

   This means that the length of a vector is unchanged after multiplying by an orthogonal matrix.

3. If $\lambda$ is a real eigenvalue of $Q$, explain why $\lambda=\pm1$.

Explain why the following statements are true.

1. If $Q$ is an orthogonal matrix, then $\det Q = \pm 1$.

2. If $Q$ is a $8\times 4$ matrix whose columns are orthonormal, then $QQ^T$ is an $8\times8$ matrix whose rank is 4.

3. If $\bhat$ is the orthogonal projection of $\bvec$ onto a subspace $W$, then $\bvec-\bhat$ is the orthogonal projection of $\bvec$ onto $W^\perp$.