Is the Kernel of a 9x10 Matrix with Rank 5 Four-Dimensional?

[teddyayalew]

Homework Statement

1. True or False: If the rank of a 9x10 matrix A is 5, then the kernel of A is 4-dimensional

Homework Equations

The Attempt at a Solution

1. I was explained that it is true because the rank and nullity of A must be 10, but I don't understand why. I understand the dimension theorem that states the sum of the nullity and rank of a transformation must equal the dimension of the domain but how is the matrix represented as a linear transformation. and what would be the domain of this transformation. I was thinking if its a 9x10 matrix T: R^10 to R^9 could a transformation and the matrix representation of T could be A. Then R^10 would be the domain and so the dimension is 10. That is why the nullity must be 5 and not 4. If I am wrong or if there is another way of approaching this problem give me your input.

Also the way i solved the problem i calculated the nullity and rank of T rather then the matrix representation of T which is = A. Is doing it as I did the same, because I feel like I am looking not at the rank and nullity of A.

 

[sunjin09]

I think what you did was right, no need to worry about the difference between A and T, they are essentially identical