Linear Chapter 2 T/F Questions

If the columns of A are linearly independent, then the columns of A span R^n.

TRUE

If the linear transformation x|--> Ax maps R^n onto R^n, then the row reduced echelon form of A is I.

TRUE

If the equation Ax=b has at least one solution for each b in R^n, then the transformation x |--> Ax is not one-to-one.

FALSE •

If A is an n x n matrix, then the equation Ax=b has at least one solution for each b in R^n.

FALSE • A needs to be invertible.

It is possible for a 4 x 4 matrix to be invertible when its columns do not span R^4.

FALSE • If the matrix is indeed invertible, then there will be a pivot position in every column and in every row. This implies no free variables and thus, the columns do indeed span R^4.

A square matrix with two identical columns can be invertible.

FALSE • This would give linear dependency, which is not defined in a proper invertible matrix.

If there is a b in R^n such that the equation Ax=b is consistent, then the solution is unique

FALSE. • Could be infinitely many solutions.

If the transpose of A is not invertible, then A is not invertible.

TRUE

If there is an n x n matrix D such that AD = I, then DA = I

TRUE

If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions.

TRUE • More vectors (columns) than rows (entries), meaning free variables and therefore infinitely many solutions.

If the columns of A span R^n, then the columns are linearly independent.

TRUE • They span R^n due to no free variables and square matrix.

If the equation Ax=0 has only the trivial solution, then A is row equivalent to the n x n identity matrix.

TRUE. • This means that it reduces to the Identity matrix, meaning a pivot in every row and every column, therefore no free variables and only the trivial solution

If an n x n matrix G cannot be row reduced to I, what can be said about the columns of G

They are linearly dependent and do not span R^n.