Question

Let M','" (F) denote the set of n × m nnatrices with entries from the field F, we define miatrix addition and multiplication of a matrix by a scalar (i.e., element of F) as you saw in Math 211. It is a fact, which you do not need to prove, that with these operations Mn m (F) is an F-vector space. Find the dimension of M,n(F). 6.

Answer 1

6. Let P and Q denote 2 arbitrary nxm matrices with entries from the field F and let α be an arbitrary scalar from the field F. Then P+Q is a nxm matrix with entries from the field F so that P+Q ∈ M_n,m(F). Hence M_n,m(F) is closed under vector addition. Also, αP is a nxm matrix with entries from the field F so that αP ∈ M_n,m(F).It implies that M_n,m(F) is closed under scalar multiplication. Further, the zero nxm matrix apparently belongs to M_n,m(F) . Hence M_n,m(F) is a vector space is an F-vector space.

Let M_ij denote the nxm matrix with 1 as the ijth entry, all other entries being zero. Then {M_ij 1≤ i≤n, 1≤j≤m} is a basis for M_n,m(F). The dimension of M_n,m(F) is, therefore, mn.