Finding orthogonal bases

Suppose that $W$ is a three-dimensional subspace of $\real^4$ with basis: 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. Expl…

Suppose that a subspace $W$ of $\real^3$ has a basis formed by

1. Find an orthogonal basis for $W$.

2. Find an orthonormal basis for $W$.

3. Find the matrix $P$ that projects vectors orthogonally onto $W$.

4. Find the orthogonal projection of $\threevec34{-2}$ onto $W$.

Find the $QR$ factorization of $A=\begin{bmatrix} 4 & 7 \\ -2 & 4 \\ 4 & 4 \end{bmatrix}$.

Consider the basis of $\real^3$ given by the vectors

1. Apply the Gram-Schmit orthogonalization algorithm to find an orthonormal basis $\uvec_1$, $\uvec_2$, $\uvec_3$ for $\real^3$.

2. If $A$ is the $3\times3$ whose columns are $\vvec_1$, $\vvec_2$, and $\vvec_3$, find the $QR$ factorization of $A$.

3. Suppose that we want to solve the equation $A\xvec=\bvec = \threevec{-9}17$, which we can rewrite as $QR\xvec = \bvec$.

   1. If we set $\yvec=R\xvec$, the equation $QR\xvec=\bvec$ becomes $Q\yvec=\bvec$. Explain how to solve the equation $Q\yvec=\bvec$ in a computationally efficient manner.

   2. Ex…

Consider the vectors and the subspace $W$ of $\real^5$ that they span.

1. Find an orthonormal basis for $W$.

2. Find the $5\times5$ matrix that projects vectors orthogonally onto $W$.

3. Find $\bhat$, the orthogonal projection of $\bvec=\fivevec{-8}3{-12}8{-4}$ onto $W$.

4. Express $\bhat$ as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$.

Consider the set of vectors

1. What happens when we apply the Gram-Schmit orthogonalization algorithm?

2. Why does the algorithm fail to produce an orthogonal basis for $\real^3$?

Suppose that $A$ is a matrix with linearly independent columns and having the factorization $A=QR$. Determine whether the following statements are true or false and explain your thinking.

1. It follows that $R=Q^TA$.

2. The matrix $R$ is invertible.

3. The product $Q^TQ$ projects vectors orthogonally onto $\col(A)$.

4. The columns of $Q$ are an orthogonal basis for $\col(A)$.

5. The orthogonal complement $\col(A)^\perp = \nul(Q^T)$.

Suppose we have the $QR$ factorization $A=QR$, where $A$ is a $7\times 4$ matrix.

1. What is the shape of the product $QQ^T$? Explain the significance of this product.

2. What is the shape of the product $Q^TQ$? Explain the significance of this product.

3. What is the shape of the matrix $R$?

4. If $R$ is a diagonal matrix, what can you say about the columns of $A$?

Suppose we have the $QR$ factorization $A=QR$ where the columns of $A$ are $\avec_1,\avec_2,\ldots,\avec_n$ and the columns of $R$ are $\rvec_1,\rvec_2,\ldots,\rvec_n$.

1. How can the matrix product $A^TA$ be expressed in terms of dot products?

2. How can the matrix product $R^TR$ be expressed in terms of dot products?

3. Explain why $A^TA=R^TR$.

4. Explain why the dot products $\avec_i\cdot\avec_j = \rvec_i\cdot\rvec_j$.