Solved on Feb 07, 2024

Find the linear operator from $P_3$ to $P_4$ where $L(p(x)) = x^2 p''(x) + p'(x) + p(0)$ .

STEP 1

Assumptions
1. $P_{3}$ denotes the space of polynomials of degree at most 3.
2. $P_{4}$ denotes the space of polynomials of degree at most 4.
3. A linear operator $L$ from $P_{3}$ to $P_{4}$ must satisfy the following properties for all $p(x), q(x) \in P_{3}$ and scalars $a, b \in \mathbb{R}$ : - Linearity: $L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))$
4. $p'(x)$ denotes the first derivative of $p(x)$ with respect to $x$ .
5. $p''(x)$ denotes the second derivative of $p(x)$ with respect to $x$ .

STEP 2

We will check each given operator to see if it satisfies the linearity properties. The operator that satisfies these properties is the linear operator we are looking for.

STEP 3

Test the first operator $L(p(x))=x^{2} p^{\prime \prime}(x) p^{\prime}(x)+p(0)$ for linearity.

STEP 4

Check if $L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))$ holds for the first operator.

STEP 5

Calculate $L(ap(x) + bq(x))$ using the first operator.
$L(ap(x) + bq(x)) = x^{2} (ap(x) + bq(x))^{\prime \prime} ((ap(x) + bq(x))^{\prime}) + (ap(x) + bq(x))(0)$

STEP 6

Observe that the expression involves products and compositions of functions, which are not linear operations. Therefore, the first operator is not linear.

STEP 7

Test the second operator $L(p(x))=p^{\prime}(x) p^{\prime \prime}(x)$ for linearity.

STEP 8

Check if $L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))$ holds for the second operator.

STEP 9

Calculate $L(ap(x) + bq(x))$ using the second operator.
$L(ap(x) + bq(x)) = (ap(x) + bq(x))^{\prime} (ap(x) + bq(x))^{\prime \prime}$

STEP 10

Observe that the expression involves products of derivatives, which are not linear operations. Therefore, the second operator is not linear.

STEP 11

Test the third operator $L(p(x))=x p(x)$ for linearity.

STEP 12

Check if $L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))$ holds for the third operator.

STEP 13

Calculate $L(ap(x) + bq(x))$ using the third operator.
$L(ap(x) + bq(x)) = x (ap(x) + bq(x))$

STEP 14

Distribute $x$ across the sum.
$L(ap(x) + bq(x)) = axp(x) + bxq(x)$

STEP 15

Observe that this expression can be written as $aL(p(x)) + bL(q(x))$ , which satisfies the linearity condition. Therefore, the third operator is linear.

STEP 16

Since we have found a linear operator, we do not need to test the remaining operators. The linear operator from $P_{3}$ to $P_{4}$ is:
$L(p(x))=x p(x)$

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Find the linear operator from $P_3$ to $P_4$ where $L(p(x)) = x^2 p''(x) + p'(x) + p(0)$ .

STEP 1

STEP 2

We will check each given operator to see if it satisfies the linearity properties. The operator that satisfies these properties is the linear operator we are looking for.

STEP 3

Test the first operator $L(p(x))=x^{2} p^{\prime \prime}(x) p^{\prime}(x)+p(0)$ for linearity.

STEP 4

Check if $L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))$ holds for the first operator.

STEP 5

Calculate $L(ap(x) + bq(x))$ using the first operator.
$L(ap(x) + bq(x)) = x^{2} (ap(x) + bq(x))^{\prime \prime} ((ap(x) + bq(x))^{\prime}) + (ap(x) + bq(x))(0)$

STEP 6

Observe that the expression involves products and compositions of functions, which are not linear operations. Therefore, the first operator is not linear.

STEP 7

Test the second operator $L(p(x))=p^{\prime}(x) p^{\prime \prime}(x)$ for linearity.

STEP 8

Check if $L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))$ holds for the second operator.

STEP 9

Calculate $L(ap(x) + bq(x))$ using the second operator.
$L(ap(x) + bq(x)) = (ap(x) + bq(x))^{\prime} (ap(x) + bq(x))^{\prime \prime}$

STEP 10

Observe that the expression involves products of derivatives, which are not linear operations. Therefore, the second operator is not linear.

STEP 11

Test the third operator $L(p(x))=x p(x)$ for linearity.

STEP 12

Check if $L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))$ holds for the third operator.

STEP 13

Calculate $L(ap(x) + bq(x))$ using the third operator.
$L(ap(x) + bq(x)) = x (ap(x) + bq(x))$

STEP 14

Distribute $x$ across the sum.
$L(ap(x) + bq(x)) = axp(x) + bxq(x)$

STEP 15

Observe that this expression can be written as $aL(p(x)) + bL(q(x))$ , which satisfies the linearity condition. Therefore, the third operator is linear.

STEP 16

Since we have found a linear operator, we do not need to test the remaining operators. The linear operator from $P_{3}$ to $P_{4}$ is:
$L(p(x))=x p(x)$

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Find the linear operator from P3P_3P3​ to P4P_4P4​ where L(p(x))=x2p′′(x)+p′(x)+p(0)L(p(x)) = x^2 p''(x) + p'(x) + p(0)L(p(x))=x2p′′(x)+p′(x)+p(0).

STEP 1

STEP 2

We will check each given operator to see if it satisfies the linearity properties. The operator that satisfies these properties is the linear operator we are looking for.

STEP 3

Test the first operator L(p(x))=x2p′′(x)p′(x)+p(0)L(p(x))=x^{2} p^{\prime \prime}(x) p^{\prime}(x)+p(0)L(p(x))=x2p′′(x)p′(x)+p(0) for linearity.

STEP 4

Check if L(ap(x)+bq(x))=aL(p(x))+bL(q(x))L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))L(ap(x)+bq(x))=aL(p(x))+bL(q(x)) holds for the first operator.

STEP 5

STEP 6

Observe that the expression involves products and compositions of functions, which are not linear operations. Therefore, the first operator is not linear.

STEP 7

Test the second operator L(p(x))=p′(x)p′′(x)L(p(x))=p^{\prime}(x) p^{\prime \prime}(x)L(p(x))=p′(x)p′′(x) for linearity.

STEP 8

Check if L(ap(x)+bq(x))=aL(p(x))+bL(q(x))L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))L(ap(x)+bq(x))=aL(p(x))+bL(q(x)) holds for the second operator.

STEP 9

Calculate L(ap(x)+bq(x))L(ap(x) + bq(x))L(ap(x)+bq(x)) using the second operator.L(ap(x)+bq(x))=(ap(x)+bq(x))′(ap(x)+bq(x))′′L(ap(x) + bq(x)) = (ap(x) + bq(x))^{\prime} (ap(x) + bq(x))^{\prime \prime}L(ap(x)+bq(x))=(ap(x)+bq(x))′(ap(x)+bq(x))′′

STEP 10

Observe that the expression involves products of derivatives, which are not linear operations. Therefore, the second operator is not linear.

STEP 11

Test the third operator L(p(x))=xp(x)L(p(x))=x p(x)L(p(x))=xp(x) for linearity.

STEP 12

Check if L(ap(x)+bq(x))=aL(p(x))+bL(q(x))L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))L(ap(x)+bq(x))=aL(p(x))+bL(q(x)) holds for the third operator.

STEP 13

Calculate L(ap(x)+bq(x))L(ap(x) + bq(x))L(ap(x)+bq(x)) using the third operator.L(ap(x)+bq(x))=x(ap(x)+bq(x))L(ap(x) + bq(x)) = x (ap(x) + bq(x))L(ap(x)+bq(x))=x(ap(x)+bq(x))

STEP 14

Distribute xxx across the sum.L(ap(x)+bq(x))=axp(x)+bxq(x)L(ap(x) + bq(x)) = axp(x) + bxq(x)L(ap(x)+bq(x))=axp(x)+bxq(x)

STEP 15

Observe that this expression can be written as aL(p(x))+bL(q(x))aL(p(x)) + bL(q(x))aL(p(x))+bL(q(x)), which satisfies the linearity condition. Therefore, the third operator is linear.

STEP 16

Since we have found a linear operator, we do not need to test the remaining operators. The linear operator from P3P_{3}P3​ to P4P_{4}P4​ is:L(p(x))=xp(x)L(p(x))=x p(x)L(p(x))=xp(x)

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Find the linear operator from $P_3$ to $P_4$ where $L(p(x)) = x^2 p''(x) + p'(x) + p(0)$ .

Test the first operator $L(p(x))=x^{2} p^{\prime \prime}(x) p^{\prime}(x)+p(0)$ for linearity.

Check if $L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))$ holds for the first operator.

Test the second operator $L(p(x))=p^{\prime}(x) p^{\prime \prime}(x)$ for linearity.

Check if $L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))$ holds for the second operator.

Calculate $L(ap(x) + bq(x))$ using the second operator.
$L(ap(x) + bq(x)) = (ap(x) + bq(x))^{\prime} (ap(x) + bq(x))^{\prime \prime}$

Test the third operator $L(p(x))=x p(x)$ for linearity.

Check if $L(ap(x) + bq(x)) = aL(p(x)) + bL(q(x))$ holds for the third operator.

Calculate $L(ap(x) + bq(x))$ using the third operator.
$L(ap(x) + bq(x)) = x (ap(x) + bq(x))$

Distribute $x$ across the sum.
$L(ap(x) + bq(x)) = axp(x) + bxq(x)$

Observe that this expression can be written as $aL(p(x)) + bL(q(x))$ , which satisfies the linearity condition. Therefore, the third operator is linear.

Since we have found a linear operator, we do not need to test the remaining operators. The linear operator from $P_{3}$ to $P_{4}$ is:
$L(p(x))=x p(x)$