91Ó°ÊÓ

Determine all vectors \(v\) that are orthogonal to \(\mathbf{u}\). $$\mathbf{u}=(0,5)$$

Prove that \(\mathbf{u} \times \mathbf{u}=\mathbf{0}\)

Prove Lagrange's Identity: \(\mathbf{u} \times \mathbf{v}\left\|^{2}=\right\| \mathbf{u}\left\|^{2}\right\| \mathbf{v} \|^{2}-(\mathbf{u} \cdot \mathbf{v})^{2}\)

Verify the triangle inequality for the vectors \(\mathbf{u}\) and \(\mathbf{v}\). $$\mathbf{u}=(1,1,1), \quad \mathbf{v}=(0,1,-2)$$

Show that the volume \(V\) of a parallelepiped having \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) as adjacent edges is \(V=|\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})|\)

Prove that the angle \(\theta\) between \(\mathbf{u}\) and \(v\) is found using \(\|\mathbf{u} \times \mathbf{v}\|=\|\mathbf{u}\|\|\mathbf{v}\| \sin \theta\).

Verify the Pythagorean Theorem for the vectors u and \(\mathbf{v}\). $$\mathbf{u}=(4,1,-5), \quad \mathbf{v}=(2,-3,1)$$

Let \(W\) be a subspace of \(R^{n}\). Prove that the intersection of \(W\) and \(W^{\perp}\) is \(\\{\boldsymbol{0}\\},\) where \(W^{\perp}\) is the subspace of \(R^{n}\) given by \(W^{\perp}=\\{\mathbf{v}: \mathbf{w} \cdot \mathbf{v}=0 \text { for every } \mathbf{w} \text { in } W\\}\).

Find bases for the four fundamental subspaces of the matrix A listed below. \(N(A)=\) nullspace of \(A \quad N\left(A^{T}\right)=\) nullspace of \(A^{T}\) \(\boldsymbol{R}(\boldsymbol{A})=\) column space of \(\boldsymbol{A} \quad \boldsymbol{R}\left(\boldsymbol{A}^{T}\right)=\) column space of \(\boldsymbol{A}^{T}\) Then show that \(N(A)=R\left(A^{T}\right)^{\perp}\) and \(N\left(A^{T}\right)=R(A)^{\perp}\). \(\left[\begin{array}{rrr}1 & 1 & -1 \\ 0 & 2 & 1 \\ 1 & 3 & 0\end{array}\right]\)

Show that \(f\) and \(g\) are orthogonal in the inner product space \(C[a, b]\) with the inner product $$\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x$$ $$\begin{aligned}&C[0, \pi], \quad f(x)=1, \quad g(x)=\cos (2 n x)\\\&n=1,2,3, \dots\end{aligned}$$