Decide if the given statement is true or false, and give a brief justification f

Question

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is talse. If T : P4 Ry is a linear transformation, then Ker(T) must be at least four-dimensional. True. This follows from the General Rank-Nullity Theorem and the fact that Rng(T) is at most 5 True. This follows from the General Rank-Nullity Theorem and the fact that Rng) is at most 4 False. Ker(T) must be at least five dimensional, since 9-4-5 False. For example, if T(ao ax a2x2 ax3 a) (, a, a2, a3, a4, 0, 0, 0, O), then KertT) False. For example, it Tau + a1x + a2x2 + a3x3·4r4)-(a04 a103, (az)2, 0, a, o, o, o, o), then Ker(7)-(0), which is 0 dimensional. (0), which is O-dimensional O

Explanation / Answer

Option (4)

Note that in option (4), the map T is a linear transformation which can easily be checked by the 2 properties :

i) T(p(x) + q(x)) = T(p(x))+T(q(x)) and

ii) T(c.p(x)) = c.T(p(x))

Where c is a scalar and p(x),q(x) are polynomials in P4

so T is a linear transformation (note that, in option 5, the map T fails these 2 properties)

So, ker(T) is by definitio, the set of all those polynomials p(x) in P4 such that, T(p(x)) = 0, where 0 is the 9-tuple zero vector in R9 .

So, (a0, a1, a2, a3, a4, a5, 0, 0, 0, 0) = (0,0,0,0,0,0,0,0,0)

implies, a0 = a1 = a2 = a3 = a4 = 0

Hence, p(x) = 0, the zero polynomial in P4

hence, the kernel is only the zero polynomial, and dim Ker (T) = 0

Hence, we get a contradiction to the assertion that Ker(T) must be of dimension atleast 4.

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