Let V R For u v V and a R define vector addition ny u v uv

Let V = R|. For u, v V| and a R| define vector addition ny u v: = u+v+3| and scalar multiplication by a u: = au+3|. It can be shown that (V,,)| is a vector space over the scalar field R|. Find the following: the sum: 3 -8 = | the scalar multiple: -2 3 = | the zero vector: = | the additive inverse of x|: x = |

Solution

As per the definition of binary operation

3+ (-8) = 3-8+3 = -2

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-2.3 = -2(3)+3(-2)-3 = -15

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u+v = u+v+3 = v if

u = -3

Hence zero vector = -3

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Inverse of x will be y such that

y+x = x+y = x+y+3 =-3

Or y = -6-x

Hence inverse of x is -6-x