91Ó°ÊÓ

Assume that \(1^{2}+2^{2}+\dots+n^{2}=a n^{3}+b n^{2}+c n+d\) Find \(a, b, c,\) and \(d .[\text { Hint: It is legitimate to use } n=0\) What is the left-hand side in that case?]

Let \(\mathbf{p}=\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right], \mathbf{q}=\left[\begin{array}{r}0 \\ 1 \\ -1\end{array}\right], \mathbf{u}=\left[\begin{array}{r}2 \\ -3 \\ 1\end{array}\right],\) and \(\mathbf{v}=\left[\begin{array}{r}0 \\ 6 \\ -1\end{array}\right]\) Show that the lines \(x=p+s u\) and \(x=q+t v\) are skew lines. Find vector equations of a pair of parallel planes, one containing each line.

Show that \(1^{3}+2^{3}+\cdots+n^{3}=(n(n+1) / 2)^{2}\)

Solve the systems of linear equations over the indicated \(\mathbb{Z}_{p}.\) \(x+2 y=1\) over \(\mathbb{Z}_{3}\) \(x+y=2\)

Solve the systems of linear equations over the indicated \(\mathbb{Z}_{p}.\) \(\begin{aligned} x+y &=1 \text { over } \mathbb{Z}_{2} \\ y+z &=0 \\ x \quad+z &=1 \end{aligned}\)

Solve the systems of linear equations over the indicated \(\mathbb{Z}_{p}.\) \(\begin{aligned} x+y &=1 \text { over } \mathbb{Z}_{3} \\ y+z &=0 \\ x \quad+z &=1 \end{aligned}\)

Solve the systems of linear equations over the indicated \(\mathbb{Z}_{p}.\) \(\begin{aligned} 3 x+2 y &=1 \text { over } \mathbb{Z}_{5} \\ x+4 y &=1 \end{aligned}\)

Solve the systems of linear equations over the indicated \(\mathbb{Z}_{p}.\) \(3 x+2 y=1\) over \(\mathbb{Z}_{7}\) \(x+4 y=1\)

When \(p\) is not prime, extra care is needed in solving a linear system (or, indeed, any equation) over \(\mathbb{Z}_{p}\) Using Gaussian elimination, solve the following system over \(\mathbb{Z}_{6} .\) What complications arise? \\[\begin{array}{l}2 x+3 y=4 \\\4 x+3 y=2\end{array}\\]