well, it depends how much you know about diagonalizable matrices. i'll assume you know nothing but this definition that an matrix with entries in a field is diagonalizable iff has a

basis such that is a diagonal matrix, where as usual is defined by

so suppose an matrix with entries in a field has distinct eigenvalues in , say let with then is a basis for

now consider the linear map defined by then and thus:

to see this we only need to show that are linearly independent. so suppose where not all of the coefficients are 0 and is minimal. call it (1). taking

of both sides of (1) gives us: and hence call this one (2). now multiply (1) by and subtract the result from (2) to get:

which contradicts the minimality of unless since are distinct, we must have

and thus by (1). contradiction! thus the elements of are linearly independent. Q.E.D.