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Questionnnnnnn a. Let V and W be vector spaces and T : V → W a...

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Basic Math

Questionnnnnnn

a. Let V and W be vector spaces and T : V → W a lineartransformation. If {T(v1), . . . T(vn)} is linearly independent inW, show that {v1, . . . vn} is linearly independent in V .

b. Define similar matrices

c Let A1, A2 and A3 be n × n matrices. Show that if A1 issimilar to A2 and A2 is similar to A3, then A1 is similar toA3.

d. Show that similar matrices have the same characteristicpolynomial and eigenvalues.

e. Determine whether the following mappings are lineartransformations.

T : V → R defined by T(x) = hx, vi, where v is a fixed nonzerovector in the real inner product space V .

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