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Problem 563

# Let $$\mathrm{V}$$ be the vector space of all polynomial functions from $$\mathrm{R}$$ into $$\mathrm{R}$$ with degree less than or equal to $$2 .$$ Find a basis for $$\mathrm{V}$$ by using the following procedures i) Find three linear functionals on $$\mathrm{V}$$ ii) Use these functionals as a basis for $$\mathrm{V}^{*}$$, the dual of $$\mathrm{V}$$. iii) Use the functionals to find a basis for $$\mathrm{V}$$.Annthilators, Transposes \& Adjoint

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To find a basis for the vector space V of all polynomial functions from R into R with degree less than or equal to 2, we first find three linear functionals $$L_1$$, $$L_2$$, and $$L_3$$ that evaluate polynomials at specific points (0, 1, and -1, respectively). Then, we use these functionals to form a basis for the dual space V* as $$\mathcal{B}^* = \{L_1, L_2, L_3\}$$. Finally, we find the annihilators for each functional and solve the resulting system of equations to obtain the basis for V: $$\mathcal{B} = \{1 + x, x^2\}$$.
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## Step 1: The given vector space V consists of all polynomial functions from R into R with degree less than or equal to 2. These polynomial functions can be written in the form $$p(x) = ax^2 + bx + c$$, where a, b, and c are real numbers. #Step 2: Find three linear functionals on V#

To find three linear functionals on V, we need to find three functions that take a polynomial from V as input and output a scalar. Let's consider the evaluation functionals, which evaluate a polynomial at a particular point. Specifically, let's consider the functionals $$L_1$$, $$L_2$$, and $$L_3$$ defined as follows: - $$L_1(p) = p(0)$$ - $$L_2(p) = p(1)$$ - $$L_3(p) = p(-1)$$ These functionals take in a polynomial $$p(x) = ax^2 + bx + c$$ and output the scalar value of the polynomial at the specific x-value chosen. These functionals are linear since the evaluation of any linear combination of polynomials is the same linear combination of their values. #Step 3: Use functionals as a basis for V*#

## Step 2: Now we want to use the functionals $$L_1, L_2$$, and $$L_3$$ as a basis for the dual space V*. Recall that the dual of a vector space consists of all linear functionals on that space. The dimension of V equals 3 since all polynomials in V have degree less than or equal to 2. Since V* has the same dimension as V, we know that having three linearly independent functionals will form a basis for V*. Given that we have three linear functionals, we can now form a basis for V* by combining them into a set: $$\mathcal{B}^* = \{L_1, L_2, L_3\}$$. #Step 4: Use functionals to find a basis for V#

Finally, we will use the functionals to find a basis for V. To do this, we will construct an annihilator for each functional and solve the resulting system of equations. An annihilator of a functional L is a polynomial p in V such that L(p) = 0. Let's find the annihilators for each of our functionals: - $$L_1$$: $$L_1(p) = p(0) = c$$, so the annihilator is a polynomial p with $$c = 0$$. - $$L_2$$: $$L_2(p) = p(1) = a + b + c$$, so the annihilator is a polynomial p with $$a + b + c = 0$$. - $$L_3$$: $$L_3(p) = p(-1) = a - b + c$$, so the annihilator is a polynomial p with $$a - b + c = 0$$. Now, let's solve the system of equations formed by the annihilators: $\begin{cases} c = 0 \\ a + b + c = 0 \\ a - b + c = 0 \end{cases}$ Solving this system, we find that $$a = 1$$, $$b = 1$$, and $$c = 0$$. Therefore, a basis for V consists of the following polynomials: $\mathcal{B} = \{1 + x, x^2\}$ In conclusion, we have found a basis for the vector space V of all polynomial functions of degree less than or equal to 2 using the procedures given: $$\mathcal{B} = \{1 + x, x^2\}$$.

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