Define vector space of real-valued functions

Question

Linear algebra questions

1. Answer the following:

a. Let U = {A∈M₂₂|det(A) = 0} (where is the vector space of 2 by 2 matrices.) Find an example that shows U is NOT a subspace of .

The matrix where U=[1 0; 0 1] the det=1. This would not be a subspace of M₂₂.

b. Let V = {ƒ∈F[0,2]|ƒ(x) = 0 for all x∈[0,1]}where is the vector space of real-valued functions defined on the interval [0, 2]. Show that is a subspace of this vector space.

1. V contains the 0 vector since

2. V is closed under scalar multiplication since u = [0,2] then cu = [0,2c]

3. V is closed under addition since ?