Which of the following are linear subspaces of the given lin

Which of the following are linear subspaces of the given linear space? For each part, either verify the two necessary properties of subspaces or briefly describe what fails. Write out basis of the subspaces, where applicable. The plane of vectors (b_1, b_2, b_3) with b_1 = b_2 in R^3. The collection of vector (b_1, b_2, b_3) with b_1

Solution

The set satisfies all the axioms of a vector space. Hence it is a subspace of R3. The set is not a subspace as the additive identity ( 0,0,0) is not in the set and also,the additive inverse of any arbitrary element ( b1,b2,b3) is not in the set. The set satisfies all the axioms of a vector space. Hence it is a subspace of M2,2 The set satisfies all the axioms of a vector space. Hence it is a subspace of M2,2