Explanation
Let W be a vector space, with an inner product . For every linear function there exists a unique vector v in W such that for all w in W. The map given by is an isomorphism. This holds for all vector spaces, and can be used to explain the Hodge dual.
Let V be an n-dimensional vector space with basis . For 0 ≤ k≤ n, consider the exterior power spaces and . For each and, we have . There is, up to a scalar, only one n-vector, namely . In other words, must be a scalar multiple of for all and .
Consider a fixed . There exists a unique linear function such that for all . This is the scalar multiple mentioned in the previous paragraph. If denotes the inner product on (n–k)-vectors, then there exists a unique (n–k)-vector, say, such that for all . This (n–k)-vector is the Hodge dual of λ, and is the image of the under the canonical isomorphism between and . Thus, .
Read more about this topic: Hodge Dual
Famous quotes containing the word explanation:
“We live between two worlds; we soar in the atmosphere; we creep upon the soil; we have the aspirations of creators and the propensities of quadrupeds. There can be but one explanation of this fact. We are passing from the animal into a higher form, and the drama of this planet is in its second act.”
—W. Winwood Reade (1838–1875)
“Auden, MacNeice, Day Lewis, I have read them all,
Hoping against hope to hear the authentic call . . .
And know the explanation I must pass is this
MYou cannot light a match on a crumbling wall.”
—Hugh MacDiarmid (1892–1978)
“Young children constantly invent new explanations to account for complex processes. And since their inventions change from week to week, furnishing the “correct” explanation is not quite so important as conveying a willingness to discuss the subject. Become an “askable parent.””
—Ruth Formanek (20th century)