# Explain true of false

1. Feb 12, 2004

### yanyin

explain why each of the following statemens is either true of false.
(a) if A is a 3X3 matrix and {v1, v2, v3} is linearly dependent set of vectors in R^3, then {Av1, Av2, Av3} is also a linearly dependent set.

(b) if A is a 3X3 invertible matrix and {v1, v2, v3} is a linearly independent set of vectors in R^3, then {Av1, Av2, Av3} is also a linearly independent set.

(c) if A is a 3X3 matrix and {v1, v2, v3} is a linearly independent set of vectors in R^3 for which {Av1, Av2, Av3} is also a linearly independent set, then the matrix A must be invertible.

(d) If A is a 3X3 matrix and {v1, v2} is a linearly independent set of vectors in R^3 for which {Av1, Av2} is also a linearly independent set, then the matrix A must be invertible.

(e) if A and B are 3X3 matrices and the product AB is known to be invertible, then it follows that B is also invertible.

2. Feb 12, 2004

### matt grime

This looks like homewrok.

a. A is linear, use this fact and the definition of Linear Dependence

b. A is linear use this fact and the definition of L. D.

c. 3 L.I. vectors in 3-d space form a basis of the space, and thus any element of the kernel would be in their span, write down such a notional element and use the hypotheses in the question

d. suppose B is not invertible, let v be non-zero and in the kernel. what is ABv?