A. Miller M340 Final May 97You may use without proof any theorem proved in class; however whenyou do so, you should state the result you are using. You may also use anyexercise, but in this case you should state the exercise and give a fullproof of the exercise in your solution.1. Let A be an m × n matrix, B an n × p matrix and a a scalar. Prove thata(AB) = (aA)B = A(aB).2. Prove thatv1, . . . , vnare linearly dependentiffv1= z or vj∈ span({v1, . . . , vj−1}) for some j with 1 < j ≤ n.3.1. For L : V → W a function from the vector space V to the vector spaceW , define L is a linear transformation.2. For L : V → W a linear transformation, define null(L).3. For L : V → W a linear transformation, define range(L).4. For L : V → W a linear transformation, define L is one-to-one.5. For L : V → W a linear transformation, define L is onto.6. For λ ∈ C and A an n × n matrix, define λ is an eigenvalue of A.7. For λ ∈ C and eigenvalue of A an n × n matrix, define X is an eigen-vector of A associated to λ.8. For A an n × n matrix, define the characteristic polynomial of A.9. For A, B two n × n matrices, define A is similar to B.10. For A an n × n matrix, define A is diagonalizable.14. Suppose Pn×nis an invertible matrix and L : MATn×n→ MATn×nisdefined by L(A) = P AP−1for any A ∈ MATn×n. Prove that1. L is a linear transformation,2. L(AB) = L(A)L(B) for every A, B ∈ MATn×n,3. L is one-to-one,4. L is onto,5. give the definition of L−1. (For this particular L not the general defi-nition of inverse)5. Suppose that A is similar to B. Prove that A and B have the samecharacteristic polynomial.Extra Credit Optional Take Home or Do-it-now Problem:Suppose A is an n × n matrix such that A2= Z. Prove thatdim(range(A)) ≤n2.Submit your solution via email to [email protected], the earlier the
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