Question

1) Define addition on R² in the usual way and define scalar multiplication by c(x1,x2)=(cx1,0). Is R² a vector space under these operations?

2) Find a nontrivial subspace of M2,2 containing no nonzero diagonal matrices.

Answer 1

No, R² is not a vector space under the given operations. A nontrivial subspace of M2,2 containing no nonzero diagonal matrices can be defined as the set of all upper triangular matrices.

Step-by-step explanation:

No, R² is not a vector space under the given operations.

For R² to be a vector space, it needs to satisfy a set of properties including closure under addition and scalar multiplication. However, in this case, scalar multiplication does not preserve the additive identity and does not satisfy the distributive property.

A nontrivial subspace of M2,2 that contains no nonzero diagonal matrices can be defined as the set of all upper triangular matrices. This subspace would consist of matrices with zeros below the main diagonal and can be represented using the {{2, a, b, c}} notation, where a, b, and c are real numbers.