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Problem 1

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

Expert verified

A field is a set F with two binary operations, addition (+) and multiplication (·), satisfying specific algebraic properties, known as field axioms. An example of an infinite field is the set of rational numbers (Q) with the usual addition and multiplication operations. An example of a finite field is the set of integers modulo a prime number, such as \(\mathbb{Z}_5\) with modulo 5 addition and multiplication operations.

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Chapter 1

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.

Chapter 1

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) \(u=1-3 t+2 t^{2}-3 t^{3}, v=-3+9 t-6 t^{2}+9 t^{3}\)

Chapter 1

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) $\mathrm{T}=\left\\{1, \mathrm{x}, \mathrm{x}^{2}\right\\} \subset \mathrm{F}[[0,1]]$

Chapter 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 \((5,3,4)\)

Chapter 1

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) \((2,-3,7),(0,0,0),(3,-1,-4)\)

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