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BaconInDistress

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I not very good at using the LaTex editor, so I took a photo of my HW questions.

View attachment 8831

For the first question, I'm not really sure how to get started, should I write out a specific case? Like what would \(\displaystyle \varphi (P)\) be when m=1?

For the second question, I know that a linear functional have two properties, one being \(\displaystyle \varphi(u +v) = \varphi(u) + \varphi(v)\) and the other one being \(\displaystyle \varphi(\lambda u) = \lambda\varphi(u)\). I'm a bit confused about the mapping, is it saying that when we have a polynomial P(z), the map makes it goes to p(0) for all z?

View attachment 8831

For the first question, I'm not really sure how to get started, should I write out a specific case? Like what would \(\displaystyle \varphi (P)\) be when m=1?

For the second question, I know that a linear functional have two properties, one being \(\displaystyle \varphi(u +v) = \varphi(u) + \varphi(v)\) and the other one being \(\displaystyle \varphi(\lambda u) = \lambda\varphi(u)\). I'm a bit confused about the mapping, is it saying that when we have a polynomial P(z), the map makes it goes to p(0) for all z?

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