SOLUTION: Let A be a 2x2 matrix. Prove that if there exists a linearly independent set of vectors {v1, v2} in R^2 such that Avi=0 for i=1,2, then A is the zero matrix. (the 'i' in Avi=0 is

Question 1036149: Let A be a 2x2 matrix. Prove that if there exists a linearly independent set of vectors {v1, v2} in R^2 such that Avi=0 for i=1,2, then A is the zero matrix.
(the 'i' in Avi=0 is a subscript as well as the '1' and '2' in v1,v2)
Thanks!
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
If the set { , } is a linearly independent set in , then it is also a basis for .
==> for any non-zero vector v in ,
for some constants and both not necessarily zero.
Left multiplying the previous equation with A, we get

, where is the zero-vector in .
In other words, the effect of left-multiplying A with any non-zero vector in is to turn it into the zero vector.
Only the 2x2 zero matrix has this effect in , and therefore .
RELATED QUESTIONS
I was given a homework sheet in class and have no idea what chapter it is from. Let A be (answered by venugopalramana)

I have no idea what this question is asking: Let B={v1, v2, v3} be a set of linearly... (answered by venugopalramana)

This is my second and last question. This is not in a textbook. a.)Let T be a linear... (answered by venugopalramana)

proofs kill me! S=(v1,v2,...vn) is a linearly independent set of vectors in the vector... (answered by venugopalramana)

Prove that if {v1, v2, v3} is a linearly independent set of vectors, then so are {v1,... (answered by solver91311,rothauserc)

Prove or show a counterexample. If v1, v2, v3 are vectors in R4 and v3 is not a linear... (answered by ikleyn)

Let v1, v2, v3 be nonzero vectors in R4 where v3 = 3·v2. Show that v1 is a linear... (answered by robertb)

Show that if s=(v1,v2,v3) is a linearly dependent set of vectors in a vector space V, and (answered by robertb)

Let v1= [0, -9, -9], v2 = [1, -3, -3], v3 = [1, 0, 0] Determine whether or not the... (answered by ikleyn)