Suppose {v1, v2, v3, . . . , vk } is a linearly independent set of vectors in n
ID: 3031683 • Letter: S
Question
Suppose {v1, v2, v3, . . . , vk } is a linearly independent set of vectors in n and suppose A is an n X n invertible matrix. Prove that {Av1, Av2, ..., Avk}, is also a linearly independent set. Hint: Use the definition of linear independence. Be careful—linear independence proofs may seem “backwards” from linear dependence proofs because you need to start with the equation for the vectors you need to show are linearly independent. Then, you need to use both of the given assumptions to solve that equation.
Explanation / Answer
Lets assmue that a1v1 +a2v2 +a3v3 ...... anvn =0
We must show that a1 = a2 = a3 ....an =0
Apply A tio above equation : A(a1v1 +a2v2 +a3v3 ...... anvn) =0
a1Av1 + a2Av2 ......anAvn =0
SInce Av1 , Av2,,,,,Avn are linealry independent, so a1 = a2=a3....an =0
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