For each given set addition operation scalar multiplication

Solution

A square matrix is called lower triangular if all the entries above the main diagonal are zero.

A vector space V is a set which is closed under finite vector addition and scalar multiplication. For a vector space, the scalars are members of a field F (say), and V is called a vector space over the field F.

If V is to be a vector space, then V must satisfy the following ten conditions for all elements (say) X, Y, Z V and any scalars p,q F:

Let us now check whether the set M of all 2x2 lower triangular matrices satisfies the above 10 conditions. Let X, Y, Z M and let the arbitrary scalars p, q F, where F is a field over which M is defined. Then

Thus M, the set of all 2x2 lower triangular matrices satisfies the above 10 conditions and hence M is a vector space.