Solve the problem.
-Determine which of the following sets is a subspace of $P _ { n }$ for an appropriate value of $n$.
A: All polynomials of the form $p ( t ) = a + b t ^ { 2 }$, where $a$ and $b$ are in $R$
B: All polynomials of degree exactly 4 , with real coefficients
C: All polynomials of degree at most 4 , with positive coefficients

A) C only
B) $A$ and B
C) A only
D) B only

Question

Solve the problem.
-Determine which of the following sets is a subspace of $P _ { n }$ for an appropriate value of $n$.
A: All polynomials of the form $p ( t ) = a + b t ^ { 2 }$, where $a$ and $b$ are in $R$
B: All polynomials of degree exactly 4 , with real coefficients
C: All polynomials of degree at most 4 , with positive coefficients

A) C only
B) $A$ and B
C) A only
D) B only

Quizplus · Accepted Answer

A) For a set to be a subspace of $P_n$, it must be closed under addition and scalar multiplication, and contain the zero vector (in this context, the zero polynomial). The set of all polynomials of the form $p(t) = a + bt^2$, where $a$ and $b$ are in $R$, satisfies these conditions for $n \geq 2$. It includes the zero polynomial (set $a = b = 0$), and the sum of any two polynomials of this form or the product of any polynomial of this form with a scalar is also of this form.B) The set of all polynomials of degree exactly 4 is not a subspace because it does not include the zero polynomial (which has degree -∞ or undefined but not 4), and the sum of two such polynomials could have a degree less than 4.C) The set of all polynomials of degree at most 4 with positive coefficients is not a subspace because it does not include the zero polynomial (its coefficients are not positive), and it is not closed under scalar multiplication (multiplying by a negative scalar would result in coefficients that are not positive).

Solve the Problem PnP _ { n }Pn​ For an Appropriate Value Of

Solve the Problem $P _ { n }$ For an Appropriate Value Of