91Ó°ÊÓ

Find the length of each of the vectors \(\vec{v}\). $$\vec{v}=\left[\begin{array}{l} 2 \\ 3 \\ 4 \end{array}\right]$$

Is \(\langle\langle A, B\rangle\rangle=\operatorname{trace}\left(A B^{T}\right)\) an inner product in \(\mathbb{R}^{n \times m} ?\) (The notation \(\langle\langle A, B\rangle\rangle\) is chosen to distinguish this product from the one considered in Example 3 and Exercise \(4 .\)

Use the formula (im \(A)^{\perp}=\operatorname{ker}\left(A^{T}\right)\) to prove the equation $$\operatorname{rank}(A)=\operatorname{rank}\left(A^{T}\right)$$

If the \(n \times n\) matrices \(A\) and \(B\) are symmetric and \(B\) is in vertible, which of the matrices must be symmetric as well? $$A+B$$

Find a basis for \(W^{\perp},\) where $$W=\operatorname{span}\left(\left[\begin{array}{l} 1 \\ 2 \\ 3 \\ 4 \end{array}\right],\left[\begin{array}{l} 5 \\ 6 \\ 7 \\ 8 \end{array}\right]\right)$$

In the space \(P_{1}\) of the polynomials of degree \(\leq 1,\) we define the inner product \\[\langle f, g\rangle=\frac{1}{2}(f(0) g(0)+f(1) g(1))\\] Find an orthonormal basis for this inner product space.In the space \(P_{1}\) of the polynomials of degree \(\leq 1,\) we define the inner product \\[\langle f, g\rangle=\frac{1}{2}(f(0) g(0)+f(1) g(1))\\] Find an orthonormal basis for this inner product space.

Find the orthogonal projection of \(\left[\begin{array}{l}49 \\ 49 \\\ 49\end{array}\right]\) onto the subspace of \(\mathbb{R}^{3}\) spanned by $$\left[\begin{array}{l} 2 \\ 3 \\ 6 \end{array}\right] \quad \text { and } \quad\left[\begin{array}{r} 3 \\ -6 \\ 2 \end{array}\right]$$

Find the least-squares solution of the system $$\begin{aligned} A \vec{x}=\vec{b}, \text { where } \quad A &=\left[\begin{array}{cc} 1 & 1 \\ 10^{-10} & 0 \\ 0 & 10^{-10} \end{array}\right] \text {and } \\ \vec{b} &=\left[\begin{array}{c} 1 \\ 10^{-10} \\ 10^{-10} \end{array}\right] \end{aligned}$$ Describe and explain the difficulties you may encounter if you use technology. Then find the solution using paper and pencil.

Show that an orthogonal transformation \(L\) from \(\mathbb{R}^{n}\) to \(\mathbb{R}^{n}\) preserves angles: The angle between two nonzero vectors \(\vec{v}\) and \(\vec{w}\) in \(\mathbb{R}^{n}\) equals the angle between \(L(\vec{v})\) and \(L(\vec{w}) .\) Conversely, is any linear transformation that preserves angles orthogonal?

Consider the orthonormal vectors \(\vec{u}_{1}, \vec{u}_{2}, \ldots, \vec{u}_{m}\) in \(\mathbb{R}^{n}\) and an arbitrary vector \(\vec{x}\) in \(\mathbb{R}^{n}\). What is the relationship between the following two quantities? $$p=\left(\vec{u}_{1} \cdot \vec{x}\right)^{2}+\left(\vec{u}_{2} \cdot \vec{x}\right)^{2}+\cdots+\left(\vec{u}_{m} \cdot \vec{x}\right)^{2} \quad \text { and } \quad\|\vec{x}\|^{2}$$ When are the two quantities equal?