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
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-> 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
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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 .
