 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2017/9/a/solution

A map between vector spaces over the same field is linear when

L (a v + b w) = a L (v) + b L (w)

for every

v, w

a, b

. Its kernel is

ker L = {v \in V : L (v) = 0}

. It contains zero because

L (0) = 0

. If

v, w \in ker L

, then

L (a v + b w) = a L (v) + b L (w) = 0.

V

. No injectivity or equality of the dimensions of

V, W

is needed.

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