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

The set \(B=\\{1, x\\}\) is a basis for the vector space \(P^{1}\) where \(\mathrm{P}^{1}\) is defined to be the vector space of all polynomials of degree less than or equal to 1 over the field of real numbers. Show that the coordinates of an arbitrary function in \(\mathrm{P}^{1}\), using the basis \(B\), are unique.

Determine whether or not \(\mathrm{u}\) and \(\mathrm{v}\) are linearly dependent if (i) \(\mathrm{u}=(3,4), \mathrm{v}=(1,-3)\) (ii) \(\mathrm{u}=(2,-3), \mathrm{v}=(6,-9)\) (iii) \(\mathrm{u}=(4,3,-2), \mathrm{v}=(2,-6,7)\) (iv) \(\mathrm{u}=(-4,6,-2), \mathrm{v}=(2,-3,1)\) (v) $\begin{array}{rrrrrr}=\mid 1 & -2 & 4 \mid, & \mathrm{v}= & 2 & -4 & 8 \\\ 13 & 0 & -1 \mid & \mid 6 & 0 & -2\end{array}$ (v) $\begin{array}{rrrrrr}1 & 2 & -3 \mid, & \mathrm{v}= & \mid 6 & -5 & 4 \\\ 16 & -5 & 4 \mid & \mid 1 & 2 & -3\end{array} \mid$ (vii) \(u=2-5 t+6 t^{2}-t^{3}, v=3+2 t-4 t^{2}+5 t^{3}\) (viii) \(u=1-3 t+2 t^{2}-3 t^{3}, v=-3+9 t-6 t^{2}+9 t^{3}\)

Find the span set of Tx where a) \(\mathrm{T}=\\{\langle 2,1>\\}\) b) \(\mathrm{T}=\\{\underline{\mathrm{OP}}, \underline{\mathrm{OQ}}\\}\) where 0 is the origin and \(\mathrm{P}, \mathrm{Q}\) and 0 are not collinear. c) $\mathrm{T}=\left\\{1, \mathrm{x}, \mathrm{x}^{2}\right\\} \subset \mathrm{F}[[0,1]]$

Which of the following vectors form a basis for \(\mathrm{R}^{3}\) ? i) \((1,1,1)\) and \((1,-1,5)\) ii) \((1,2,3),(1,0,-1),(3,-1,0)\) and \((2,1,-2)\) iii) \((1,1,1) \quad(1,2,3)\) and \((2,-1,1)\) iv) \((1,1,2),(1,2,5)\) and \((5,3,4)\)

Determine whether or not the following vectors in \(\mathrm{R}^{3}\) are linearly dependent: i) \((1,-2,1),(2,1,-1),(7,-4,1)\) ii) \((1,-3,7),(2,0,-6),(3,-1,-1),(2,4,-5)\) iii) \((1,2,-3),(1,-3,2),(2,-1,5)\) iv) \((2,-3,7),(0,0,0),(3,-1,-4)\)