Question

et A be an n by n matrix, with real valued entries. Suppose that A is NOT invertible.
Which of the following statements are true?
Select ALL correct answers.
Note: three submissions are allowed for this question.

The columns of A are linearly independent.The linear transformation given by A is not one-to-one.The columns of A span Rⁿ.The linear transformation given by A is onto Rⁿ.There is an n by n matrix D such that AD=I_n.None of the above.

Answer 1

. The columns of A are linearly independent. FALSE: If A is not invertible, then the columns of A are linearly dependent.

2. The linear transformation given by A is not one-to-one. TRUE: The linear transformation given by A is one-to-one if and only if A is invertible.

3. The columns of A span Rⁿ. FALSE: The columns of A will span Rⁿ if and only these columns are linearly independent, in which case, A will be invertible.

4. The linear transformation given by A is onto Rⁿ. There is an n x n matrix D such that AD=I_n. TRUE: If the linear transformation given by A is onto, then the columns of A span Rⁿ. Then A is invertible so that there is an n x n matrix D such that AD=I_n.