Using the facts that ϕ0 and ϕ1 are orthogonal to ϕn for n>1, we have ∫a^b(αx+β) ϕn(x) d x =α

step 1 So the question says verify by direct integration that function are orthogonal with respect to i

Using the facts that ϕ0 and ϕ1 are orthogonal to ϕn for n>1,...

Using the facts that $\phi_{0}$ and $\phi_{1}$ are orthogonal to $\phi_{n}$ for $n>1$, we have $$\begin{aligned} \int_{a}^{b}(\alpha x+\beta) \phi_{n}(x) d x &=\alpha \int_{a}^{b} x \phi_{n}(x) d x+\beta \int_{a}^{b} 1 \cdot \phi_{n}(x) d x \\ &=\alpha \int_{a}^{b} \phi_{1}(x) \phi_{n}(x) d x+\beta \int_{a}^{b} \phi_{0}(x) \phi_{n}(x) d x \\ &=\alpha \cdot 0+\beta \cdot 0=0 \end{aligned}$$ for $n=2,3,4, \dots$

Using the facts that $\phi_{0}$ and $\phi_{1}$ are orthogonal to $\phi_{n}$ for $n>1$, we have
$$\begin{aligned}
\int_{a}^{b}(\alpha x+\beta) \phi_{n}(x) d x &=\alpha \int_{a}^{b} x \phi_{n}(x) d x+\beta \int_{a}^{b} 1 \cdot \phi_{n}(x) d x \\
&=\alpha \int_{a}^{b} \phi_{1}(x) \phi_{n}(x) d x+\beta \int_{a}^{b} \phi_{0}(x) \phi_{n}(x) d x \\
&=\alpha \cdot 0+\beta \cdot 0=0
\end{aligned}$$
for $n=2,3,4, \dots$

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Key Concepts

Linearity of Integration

The linearity of integration is a fundamental property stating that the integral of a sum of functions is the sum of their integrals, and the integral of a constant multiplied by a function is equal to the constant times the integral of the function. This property enables decomposing integrals of linear combinations of functions into simpler, more manageable parts, especially in the context of orthogonal functions.

Orthogonality

Orthogonality in function spaces refers to the concept where two functions have an inner product (often defined as an integral over a given interval) equal to zero. This property is fundamental in many areas of mathematics since it implies that the functions are independent in some sense and can serve as a basis for expressing other functions.

Inner Product Spaces

An inner product space is a vector space endowed with an additional structure called an inner product, which allows the measurement of angles and lengths. In the context of functions, this inner product is commonly defined as an integral of the product of two functions over a certain interval, and it is key to defining concepts such as orthogonality and norm.

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