Linear Algebra Exam 1 Review

Every matrix is row equivalent to a unique matrix in echelon form.

False

Reduced row-echelon form (RREF) is unique for every matrix.

Any system of n linear equations in n variables has at most n solutions.

False

If a system of linear equations has two different solutions, it must have infinitely many solutions.

True

If a system of linear equations has no free variables, then it has a unique solution.

False

If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax=b and Cx=d have exactly the same solution sets.

True

If a system Ax=b has more than one solution, then so does the system Ax=0.

True

If A is an mxn matrix and the equation Ax=b is consistent for some b, then the columns of A span R^m.

False

If an augmented matrix [A,b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax=b is consistent.

False

If matrices A and B are row equivalent, then they have the same reduced echelon form.

True

The equation Ax=0 has the trivial solution if and only if there are no free variables.

False

If the equation Ax = 0 has the trivial solution it does not imply that there are no free variables.

If A is an mxn matrix and the equation Ax=b is consistent for every b in R^m, then A has m pivot columns.

True

If an mxn matrix A has a pivot position in every row, then the equation Ax=b has a unique solution for each b in R^m.

True

If an nxn matrix A has n pivot positions, then the reduced echelon form of A is the nxn identity matrix.

False

If 3x3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations.

True

If A is an mxn matrix, if the equation Ax=b has at least two different solutions, and if the equation Ax=c is consistent, then Ax=c has many solutions.

True

If A and B are row equivalent mxn matrices and if the columns of A span R^m, then so do the columns of B.

True

If none of the vectors in the set S={v1,v2,v3} in R^3 is a multiple of one of the other vectors, then S is linearly independent.

False

If {u,v,w} is linearly independent, then u,v and w are not in R^2.

True

In some cases, it is possible for four vectors to span R^m.

False

If u and v are in R^m, then -u is in span {u,v}.

True

If u,v, and w are nonzero vectors in R^2, then w is a linear combination of u and v.

False

If w is a linear combination of u and v in T:R^n, then u is a linear combination of v and w.

False

Suppose that v1,v2, and v3 are in R^5, v2 is not a multiple of v1 and v3 is not a linear combination of v1 and v2. Then {v1,v2,v3} is linearly dependent.

False

A linear transformation is a function.

True

If A is a 6x5 matrix, the linear transformation x->Ax cannot map R^5 onto R^6.

True

If A is an mxn matrix with m pivot columns, then the linear transformation x->Ax is a one-to-one mapping.

False

For the transformation to be one-to-one, there must be a pivot in every column. Since A has n columns and m pivots, m might be less than n.