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b3.4_5.1_1.1____2.1.1. title: The result of the inner product must be greater than zero if the vector is not zero. This can be related to the length of a vector and is very useful when trying to understand abstract vector spaces.

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The result of the inner product must be greater than zero if the vector is not zero.
vβ‰ 0β‡’βŸ¨v,v⟩>0\mathrm{v \neq 0 \Rightarrow \lang{v,v}\rang > 0}
This condition for a inner product can be related to the length of a vector(from2) and is very useful when trying to understand abstract vector spaces.
Example 1. Let E=R2[x]E = \mathbb{R}_2[x](from4) and the mapping EΓ—E↦RE \times E \mapsto \mathbb{R} is defined by:
⟨P,G⟩=∫0+∞P(x)Q(x)eβˆ’xdx\lang{P, G}\rang = \int^{+\infin}_0P(x)Q(x)e^{-x}dx
How can we show ⟨P,G⟩\lang{P, G}\rang is an inner product over EE? If you are just understanding the inner product as β€˜dot product’ which is called β€˜standard inner product’, it might be difficult to see how we can interpret as an inner product. One of the conditions(from1) which I just mentioned in this document is particularly challenging to apply.
Pβ‰ 0β‡’βŸ¨P,P⟩>0\mathrm{P \neq 0 \Rightarrow \lang{P,P}\rang > 0}
Check the β€˜custom’ inner product ⟨P,G⟩\lang{P, G}\rang permits the condition.
⟨P,P⟩=∫0+∞P(x)P(x)eβˆ’xdx\lang{P, P}\rang = \int^{+\infin}_0P(x)P(x)e^{-x}dx
=∫0+∞(P(x))2eβˆ’xdx>0= \int^{+\infin}_0(P(x))^2e^{-x}dx > 0
eβˆ’xe^{-x} is greater than 00 over [0,+∞)[0, +\infin) and (P(x))2(P(x))^2 is greater than 00 if P(x)β‰ 0P(x) \neq 0 over [0,+∞)[0, +\infin), ⟨P,G⟩\lang{P, G}\rang could be an inner product.
Example 2. Let EE be the vector space of the continuous functions on [0,1][0, 1], together with the inner product.
⟨f,g⟩=∫01f(t)g(t)dt\lang{f, g}\rang = \int^1_0f(t)g(t)dt
Consider the linear subspace FβŠ‚EF \subset E, how can we show this?
FβŠ₯={0E}F^\perp =\{0_E\}
Let’s rewrite the FβŠ₯F^\perp(from3).
FβŠ₯={g∈E∣⟨f,g⟩=0,βˆ€f∈F}F^\perp =\{g \in E | \lang{f,g}\rang=0, {\forall}f \in F\}
Then we might see:
1.
If f(x)=0f(x) = 0 over [0,1][0, 1], gg could be anything (include 0E0_E)
2.
If f(x)β‰ 0f(x) β‰  0 over [0,1][0, 1], gg must be 0E0_E
To permit both conditions 1 and 2, FβŠ₯F^\perp should be:
FβŠ₯={0E}F^\perp =\{0_E\}
parse me : μ–Έμ  κ°€ 이 글에 쓰이면 쒋을 것 같은 μž¬λ£Œλ“€.
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from : 과거의 μ–΄λ–€ 생각이 이 생각을 λ§Œλ“€μ—ˆλŠ”κ°€?
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supplementary : μ–΄λ–€ μƒˆλ‘œμš΄ 생각이 이 λ¬Έμ„œμ— μž‘μ„±λœ 생각을 λ’·λ°›μΉ¨ν•˜λŠ”κ°€?
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opposite : μ–΄λ–€ μƒˆλ‘œμš΄ 생각이 이 λ¬Έμ„œμ— μž‘μ„±λœ 생각과 λŒ€μ‘°λ˜λŠ”κ°€?
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to : 이 λ¬Έμ„œμ— μž‘μ„±λœ 생각이 μ–΄λ–€ μƒκ°μœΌλ‘œ λ°œμ „λ˜κ³  μ΄μ–΄μ§€λŠ”κ°€?
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참고 : 레퍼런슀
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