Problem 7
Consider the linear system \(3 x+y+4 z=3\). In each case solve the system by \(4 x+3 y+z=1\) reducing the augmented matrix to reduced row-echelon form over the given field: a. \(\mathbb{Z}_{5}\) b. \(\mathbb{Z}_{7}\)
Problem 8
In each case, find a unitary matrix \(U\) such that \(U^{H} A U\) is diagonal. a. \(A=\left[\begin{array}{rr}1 & i \\ -i & 1\end{array}\right]\) b. \(A=\left[\begin{array}{cc}4 & 3-i \\ 3+i & 1\end{array}\right]\) c. \(A=\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right] ; a, b,\) real d. \(A=\left[\begin{array}{cc}2 & 1+i \\ 1-i & 3\end{array}\right]\) e. \(A=\left[\begin{array}{ccc}1 & 0 & 1+i \\ 0 & 2 & 0 \\ 1-i & 0 & 0\end{array}\right]\) f. \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 1+i \\ 0 & 1-i & 2\end{array}\right]\)
Problem 8
Let \(A_{0}\) be formed from \(A\) by deleting rows 2 and 4 and deleting columns 2 and \(4 .\) If \(A\) is positive definite, show that \(A_{0}\) is positive definite.
Problem 8
Let \(K\) be a vector space over \(\mathbb{Z}_{2}\) with basis \(\\{1, t\\},\) so \(K=\left\\{a+b t \mid a, b,\right.\) in \(\left.\mathbb{Z}_{2}\right\\} .\) It is known that \(K\) becomes a field of four elements if we define \(t^{2}=1+t\) Write down the multiplication table of \(K\).
Problem 8
Let \(A^{-1}=A=A^{T}\) where \(A\) is \(n \times n\) Given any orthogonal \(n \times n\) matrix \(U\), find an orthogonal matrix \(V\) such that \(A=U \Sigma_{A} V^{T}\) is an SVD for \(A\). If \(A=\left[\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right]\) do this for: a. \(U=\frac{1}{5}\left[\begin{array}{rr}3 & -4 \\ 4 & 3\end{array}\right]\) b. \(U=\frac{1}{\sqrt{2}}\left[\begin{array}{rr}1 & -1 \\ 1 & 1\end{array}\right]\)
Problem 9
If \(A\) is positive definite, show that \(A=C C^{T}\) where \(C\) has orthogonal columns.
Problem 9
Show that \(\langle A \mathbf{x}, \mathbf{y}\rangle=\left\langle\mathbf{x}, A^{H} \mathbf{y}\right\rangle\) holds for all \(n \times n\) matrices \(A\) and for all \(n\) -tuples \(\mathbf{x}\) and \(\mathbf{y}\) in \(\mathbb{C}^{n}\).
Problem 9
Let \(K\) be a vector space over \(\mathbb{Z}_{3}\) with basis \(\\{1, t\\},\) so \(K=\left\\{a+b t \mid a, b,\right.\) in \(\left.\mathbb{Z}_{3}\right\\} .\) It is known that \(K\) becomes a field of nine elements if we define \(t^{2}=-1\) in \(\mathbb{Z}_{3} .\) In each case find the inverse of the element \(x\) of \(K\) : a. \(x=1+2 t\) b. \(x=1+t\)
Problem 9
Let \(U\) be a subspace of \(\mathbb{R}^{n}\) a. Show that \(U^{\perp}=\mathbb{R}^{n}\) if and only if \(U=\\{\mathbf{0}\\}\). b. Show that \(U^{\perp}=\\{\mathbf{0}\\}\) if and only if \(U=\mathbb{R}^{n}\).
Problem 9
Find a SVD for the following matrices: a. \(A=\left[\begin{array}{rr}1 & -1 \\ 0 & 1 \\ 1 & 0\end{array}\right]\) b. \(\left[\begin{array}{rrr}1 & 1 & 1 \\ -1 & 0 & -2 \\ 1 & 2 & 0\end{array}\right]\)
