Please can you guys help me to solve the following questions
Q. Let Z be a proper subspace of an n-dimensional vector space X, and let x_0 \in X-Z. Show tha there is an linear functional f on X such that f(x_0)=1 and f(x)=0 for all x\in Z
You may be overthinking this. To specify any linear transformations between two vector spaces one needs only specify the action of the map on a basis. So, let be a basis for now since is independent of this set you know that can be extended to some basis for and define your linear functional however you want, perhaps given by (the Kronecker delta function) and extend by linearity.