Is A invertible if {v1, v2} and {Av1, Av2} are linearly independent sets?

[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.

 

[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?

 

[M98Ranger]

(a) This statement is false. If {v1, v2, v3} is linearly dependent, it means that at least one of the vectors can be written as a linear combination of the others. This means that at least one of the vectors in {Av1, Av2, Av3} can also be written as a linear combination of the others, making the set linearly dependent.

(b) This statement is true. If A is an invertible matrix, it means that it has a unique inverse. This means that for any vector x in R^3, there exists a unique vector y in R^3 such that Ax=y. If {v1, v2, v3} is linearly independent, it means that none of the vectors can be written as a linear combination of the others. This property carries over to {Av1, Av2, Av3}, making it linearly independent as well.

(c) This statement is true. If {v1, v2, v3} is linearly independent and {Av1, Av2, Av3} is also linearly independent, it means that A has a unique inverse. This is because if A did not have a unique inverse, it would mean that there exists a non-zero vector x such that Ax=0, which would contradict the linear independence of {v1, v2, v3}.

(d) This statement is false. Similar to part (a), if {v1, v2} is linearly independent, it does not necessarily mean that {Av1, Av2} is also linearly independent. This depends on the specific values of A and the vectors v1 and v2.

(e) This statement is true. If AB is invertible, it means that it has a unique inverse. This means that there exists a unique matrix C such that (AB)C=I, where I is the identity matrix. This implies that B must also have a unique inverse, as it can be written as (AB)C=C^-1, where C^-1 is the inverse of C.