91Ó°ÊÓ

Define a field and give an example of i) an infinite field and ii) a finite field.

Show that the set of semi-magic squares of order \(3 \times 3\) form a vector space over the field of real numbers with addition defined as: \(\mathrm{a}_{\mathrm{ij}}+\mathrm{b}_{\mathrm{ij}}=(\mathrm{a}+\mathrm{b})_{\mathrm{ij}}\) for \(\mathrm{i}, \mathrm{j}=1, \ldots, 3\).

A force of 25 newtons is being opposed by a force of 20 newtons, the acute angle between their lines of action being \(60^{\circ}\). Use a scale diagram to approximate the magnitude and direction of the resultant force.

Find the norm of the three dimensional vector \(\mathrm{u}=(-3,2,1)\) and the distance between the points \((-3,2,1)\) and \((4,-3,1)\).

The standard basis for \(R^{3}\) is $$ \begin{aligned} \mathrm{B}_{\mathrm{s}}=\\{&|1||0||0|\\} \\ \\{&|0||1||0|\\} \\ \\{&|0||0||1|\\} \end{aligned} $$

Let W be the set consisting of all \(2 \times 3\) matrices of the form $$ \left|\begin{array}{lll} a & b & 0 \\ 0 & c & d \end{array}\right| $$ where a, \(b, c, d\) are real numbers. (1) Show that \(\mathrm{W}\) is a subspace of \(\mathrm{V}\), the set of all \(2 \times 3\) matrices under the operation of addition over the field of real numbers.