Linear Algebra and ItsApplications III, Lay, p. 223, section 4.1, question 5 The

Question

Linear Algebra and ItsApplications III, Lay, p. 223, section 4.1, question 5 The cramster solution provides for three of the requirementsfor a given set to be a subspace of Pn: closed under vectoraddition, and closed under scalar multiplicaiton. However, noreference is made to the zero vector of Pn being inp(t)=a*t^2. How can I show the zero vector of Pn being is in p(t)=a*t^2? Linear Algebra and ItsApplications III, Lay, p. 223, section 4.1, question 5 The cramster solution provides for three of the requirementsfor a given set to be a subspace of Pn: closed under vectoraddition, and closed under scalar multiplicaiton. However, noreference is made to the zero vector of Pn being inp(t)=a*t^2. How can I show the zero vector of Pn being is in p(t)=a*t^2?

Explanation / Answer

The zero polynomial occurs when all the coefficients of apolynomial are zero.
p(t) = a* t^2 when a = 0 p(t) = 0 * t^2 = 0..which is the definition of the zeropolynomial.
So it is a subspace of Pn .. sinceyou already verified the other two conditionsrequired.

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