Question

Consider the matrix A where a, b, c are real numbers. In this problem, we consider what the ca cb dimensions of subspaces associated to A could be; these answers may depend on the values of a, b, and c, and we want to see what the possible dimensions are. (a) Could the null space of A be 0-dimensional? 1-dimensional? 2-dimensional? (b) Could the column space of A be 0-dimensional? 1-dimensional? 2-dimensional?

Answer 1

A = [a b ca cb] to find the dimension of nullspace first we change the matrix to row reduced echelon form [a b ca cb] R_2 rightarrow R_2 - cR_1 ~ [a b 0 cb - cb] ~ [a b 0 0] therefore the matrix A has one zero row therefore x it has nullspace contains 1- Dimensional if a notequalto 0, b notequalto 0 * nullspace of A be 2-dimensional if a = 0, b = 0 for dimension of the column space is the rank of the matrix Rank of A is number of nonzero rows in the echelon form, 2n part(a) we have [a b 0 0] * column space of A has 1-dimensional is atleast one of a notequalto 0, b notequalto 0 * column space of A has 0-dimensional if a = 0 and b = 0 * 2 dimensional never possible for columnspace