Let B v1vn be a basis of a vector space V and let T V V b

Let B = {v_1,...,v_n} be a basis of a vector space V and let T : V -> V be a linear transformation with the matrix [T]_B. Prove that T is one-to-one and onto if and only if det[T]_B != 0

Solution

T : V V is a linear transform with standard matrix [T]_B. Then,

1 T is onto if and only if the columns of [T]_B span V .

2 T is one-to-one if and only if the columns of [T]_B are linearly independent.

The following conditions are equivalent for a n × n matrix A

If det [T]_B    = 0, then [T]_B is not invertible and the columns of [T]_B are linearly dependent. And do not form a basis for V. Then T is not one-to-one or onto. Conversely, if T is not one-to-one and onto, then the columns of [T]_B are linearly dependent, then det [T]_B = 0