00:02
Let v be an n -dimensional vector space over a field f with vectors b1, b2 up to bn in v, such that the set b formed by those vectors b1, b2 up to bn forms the basis for the vector space v.
00:22
We define a function t from vector space v to field f to the n, that is the n -tuples of elements in v, in f sorry, in the following way.
00:36
If v is a vector in v, it can be written as a linear combination of the vectors in the base b as v equal a1 times b1 plus a2 times b2 plus up to an times bn for a1, a2 up to an in the field f, then t of v is a1, a2 up to an.
00:58
There is a transformation of any vector in vector space v is the vector of coordinates of that given vector v in the base b of the vector space v.
01:14
So we prove that t is linear transformation and in part b we determine whether t is one -to -one and or onto and prove the results.
01:23
So in part a we prove two things.
01:28
First we prove that if we take two vectors u and v in vector space v, we're going to prove that the function t at the sum of u plus v is equal to t at u plus t at v.
01:54
So let us prove that t of u plus v equal t of u plus t of v.
02:10
So we start by saying that since u and v are vectors in v, then there exists let's say a1, a2 up to an and c1, c2 up to cn numbers or values...