V = R[x]3, W = R[x]6 be the vector spaces consisting of zero and polynomials in z of degree <3 and degree ≤6 respectively. This question concerns the map a: V→ W which sends a polynomial f(x) in V to f(x²) in W.

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A: Note:  As you requested only Part C(I) so I answered here only that.

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Q: Prove that V is a vector space with the operations

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Q: 13. Prove that if V and W are vector spaces and T: V → W is a linear transformation then Range(T) is…

A: Subspace of a vector space

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Q: 2: Show that a map T is linear if and only if T(Au + v) = XT(u)+ T(v) for all A ER and vectors u, v.

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Q: Show that V={(x,2020x): x in R} is a vector space

A: According to the given information, it is required to show that the given set is a vector space.

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Q: (a) ]Find all vectors in the form x= X2 x3 such that T(x) is a scalar multiple of x.

A: I am solving the first part for you.The given linear transformation is : from To find all vectors…

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Q: et V be the set of ordered pairs (a, b) of real numbers with Idition in V and scalar multiplication…

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