Recall that we say a linear transformation T Rn rightarrow R

Recall that we say a linear transformation T: R^n rightarrow R^m is one-to-one if distinct inputs have distinct outputs (i.e. if V_1 notequalto v_2, then T (v_1) notequalto T(v_2)). We say T is onto if for every vector b in R^m we have b = T(upsilon) for some vector upsilon in R^n. Let A be a 6 times 4 matrix. Suppose rank(A)=4. Is A one-to-one? Explain your reasoning. Is A onto? Explain your reasoning. What is the nullity of A^T? Explain your reasoning.

Solution

We know that if T: RnRm is a linear transformation and if A is the standard matrix of T , then T is one-to-one only if the columns of A are linearly independent. Here, A is a 6x4 matrix with rank 4. Therefore, dim(Col(A) ) = 4, i.e. the columns of A are linearly independent. Hence T is one-to-one. We know that if T: RnRm is a linear transformation and if A is the standard matrix of T , then T is onto only if the columns of A span Rm. Here, A is a 6 x 4 matrix with rank 4. Hence the columns of A are linearly independent and ,therefore, the columns of A span R4. Therefore, T is onto. We know that the rank of AT is same as that of A, i.e. the rank of AT is 4. Since AT is a 4x6 matrix, hence as per the rank-nullity theorem, the nullity of AT is 6-4 = 2.