Question

Suppose A is an m × n matrix in which m> n. Suppose also that the rank of A equals n Show that A is one to one. Hint: If not, there exists a vector x such that Ax = 0, and this implies at least one column of A is a linear combination of the others. Show this would require the rank to be less than n.

Answer 1

Explanation:

If T:Rn→Rm is a linear transformation and if A is the standard matrix of T, then the following are equivalent:

1. T is one-to-one.

2. T(x) = 0 has only the trivial solution x=0.

3. If A is the standard matrix of T, then the columns of A are linearly independent.

Here, A is a mxn matrix where m ≥ n and the rank of A equals n. It implies that the columns of A are linearly independent, for, otherwise, the rank of A would be less than n. Hence the linear transformation represented by A is one-to-one.