Question

Determine if the statement is true or false, and justify your answer. If A is a matrix with linearly independent columns, then Ax = 0 has nontrivial solutions.

Answer 1

The statement is false because if a matrix has linearly independent columns, the only solution to the equation Ax = 0 is the trivial solution, where x is the zero vector.

Step-by-step explanation:

The statement "If A is a matrix with linearly independent columns, then Ax = 0 has nontrivial solutions" is false. In linear algebra, if a matrix A has linearly independent columns, it means that no column can be written as a linear combination of the other columns.

This implies that the only solution to the equation Ax = 0 is the trivial solution, where x is the zero vector. The reason is that the columns of A span a space of dimension equal to the number of columns, and if those columns are linearly independent, the null space of A (the set of all solutions to Ax = 0) can only contain the zero vector.