math linear algebra Is it true or false 2 5 points each For

(2) (5 points each) For each of the following statements, determine if it is true or false. Justify your answer. (a) Let B be a 6 x 8 matrix. If the transformation z Br is onto, then dim Nul(B) 2.

Solution

(a) We know that T is onto if and only if the equation T(x)=b has solutions for every b i.e. if the equation Bx = b has solutions for every b . This means that the equation Bx = 0 has only a trivial solution. This means that there are no free variables, which in turn means that the rank of B is 6 and the nullity of B is 2.

(b) Here, C is a 5x4 matrix such that dim Null(C) = 0. Then the rank of C is 4 -0 = 4. Since the row and the column ranks of a matrix are equal, dim Col (C) = 4 and hence Col(C) = R⁴.