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Example 1

Let \(t = \begin{bmatrix} 0 \\ 1 \\ 2\end{bmatrix}+ 2\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}\) be a \(3\)-tuple with entries from \(\mathbb{R}\). What is \(t_2^2\)?

Example 2

Write out all the elements of the set \(\{n \in \mathbb{N} : (n-2)^2 \leq 10\}\).

Example 3

Write out all the elements of the set \(\left\{\begin{bmatrix} a \\ b \end{bmatrix} \in \mathbb{Z}^2 : |a+b|=1,~a\geq 0,~b \geq -1\right\}\).

Example 4

Evaluate the expression \(1 + (1 + (1 + 1)\cdot 1)\) over \(GF(2)\).

Example 5

Give the multiplicative inverse of \(4-\sqrt{5}\) in the form \(a+b\sqrt{5}\) where \(a,b \in \mathbb{Q}\).

Example 6

Let \(a \in \mathbb{F}\) where \(\mathbb{F}\) denotes a field. Suppose that \(a \neq 0\). Show that \(a^{-k} = \left(a^{-1}\right)^k\) for every natural number \(k\). (You may use the fact that \(a^k b^k = (ab)^k\) for all \(a,b \in \mathbb{F})\).

Example 7

Let \(\mathbb{Q}(\sqrt{2})\) denote the set of numbers of the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are rational numbers. Show that \(\mathbb{Q}(\sqrt{2})\) under the usual addition and multiplication for real numbers form a field.

Example 8

Consider the function \(T:\mathbb{R}^2 \rightarrow \mathbb{R}^2\) given by \[ T\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 2x - y \\ x + y \end{bmatrix}.\] Show that \(T\) is surjective.