Prove that, for m n matrices A and B, rank (A B) rank(A) rank(B).
MTH 132 Lecture 2 Limits General definition ● Let f(x൦) be a function near a. (a may not be in the domain) ● We say that the limit of f(x൦) = L as x approaches a is denoted by limit f(x൦) = L ● If f(x൦) is arbitrarily close to L by x sufficiently close to a (but not equal to) = the limit. ○ We are interested in the behaviour of f(x൦) near a. ○ Finding the limit has nothing to do with the value f(a). Both Sides ● For a limit to be defined ‘normally’ it has to have the same solution from both sides; approaching from the positive side and approaching from the negative side. ○ Otherwise it does not exist. ○ Some can non exist because of oscillation. ○ . ● We can also define a one sided limit, where instead of approaching from both sides like limit x approaches 1, ○ we add a plus symbol to designate whether we’re approaching from the positive side (the right) ○ or a minus symbol to say we’re approaching from the negative side (the left). ● If the limit exists from both sides, then the solution with the + and the solution with the would be equivalent. Otherwise, if they’re different then the limit does not exist. Finding limits! MTH 132 Lecture 3 Approaching L Review of previous concepts ● Limit x→a f