True/False 1) The nullspace of a 3x4 matrix cannot consist of only the zero vect

Question

True/False

1) The nullspace of a 3x4 matrix cannot consist of only the zero vector.

2) The nullspace of a 4x3 matrix cannot consist of only the zero vector.

3) The set of all vectors of the form 1

                                                          x

                                                          y

where x and y range over all real numbers, is a subspace of R^3.

4) 3) The set of all vectors of the form 0

                                                              x

                                                              y

where x and y range over all real numbers, is a subspace of R^3.

Explanation / Answer

correct answer with explanations:

1) T {matrix maps R^4 to R^3}

2) F {counterexample: use invertible 3X3 matrix and add 4th row that is copy of third}

3) F {does not contain 0 vector}

4) T {= span of [0 1 0]^T and [0 0 1]^T}

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