91Ó°ÊÓ

If \(A\) is an \(m \times n\) matrix of rank \(r,\) what are the dimensions of \(N(A)\) and \(N\left(A^{T}\right) ?\) Explain.

Let \(A\) be an \(m \times n\) matrix of rank \(n\) and let \(P=\) \(A\left(A^{T} A\right)^{-1} A^{T}\) (a) Show that \(P \mathbf{b}=\mathbf{b}\) for every \(\mathbf{b} \in R(A)\). Explain this property in terms of projections. (b) If \(\mathbf{b} \in R(A)^{\perp},\) show that \(P \mathbf{b}=\mathbf{0}\) (c) Give a geometric illustration of parts (a) and (b) if \(R(A)\) is a plane through the origin in \(\mathbb{R}^{3}\)

Write out the Fourier matrix \(F_{8} .\) Show that \(F_{8} P_{8}\) can be partitioned into block form: $$\left(\begin{array}{cc} F_{4} & D_{4} F_{4} \\ F_{4} & -D_{4} F_{4} \end{array}\right)$$

Let \(A\) be an \(8 \times 5\) matrix of \(\operatorname{rank} 3,\) and let \(\mathbf{b}\) be a nonzero vector in \(N\left(A^{T}\right)\) (a) Show that the system \(A \mathbf{x}=\mathbf{b}\) must be inconsistent. (b) How many least squares solutions will the system \(A \mathbf{x}=\mathbf{b}\) have? Explain.

Find the distance from the point (1,1,1) to the plane \(2 x+2 y+z=0\)

Prove that the transpose of an orthogonal matrix is an orthogonal matrix.

Find the distance from the point (2,1,-2) to the plane $$6(x-1)+2(y-3)+3(z+4)=0$$

If \(Q\) is an \(n \times n\) orthogonal matrix and \(\mathbf{x}\) and y are nonzero vectors in \(\mathbb{R}^{n},\) then how does the angle between \(Q \mathbf{x}\) and \(Q \mathbf{y}\) compare with the angle between \(\mathbf{x}\) and \(\mathbf{y} ?\) Prove your answer.

Show that if $$\left(\begin{array}{cc} A & I \\ O & A^{T} \end{array}\right)\left(\begin{array}{l} \hat{\mathbf{x}} \\ \mathbf{r} \end{array}\right)=\left(\begin{array}{l} \mathbf{b} \\ \mathbf{0} \end{array}\right)$$ then \(\hat{\mathbf{x}}\) is a least squares solution of the system \(A \mathbf{x}=\mathbf{b}\) and \(\mathbf{r}\) is the residual vector.

Use the zeros of the Legendre polynomial \(P_{2}(x)\) to obtain a two-point quadrature formula $$\int_{-1}^{1} f(x) d x \approx A_{1} f\left(x_{1}\right)+A_{2} f\left(x_{2}\right)$$