page2058 Vector Spaces Matthias Lorentzen...mattegrisenforlag.com
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Vector Spaces
Introduction: An isomorphism is a crucial concept in mathematics. It is the formal way of saying two things are "the same" , even if they are wearing different clothes. For the journey toward Lie Algebra , isomorphisms are what allow us to say that the Lie Algebra of rotations in `bbb"3D"` is "the same" as the Lie Algebra of `3 times 3` antisymmetric matrices.
Exercise: Define the conditions under which a mapping (function) `phi : bbb"V" rightarrow bbb"W"` between two vector spaces over the same field `bbb"F"` constitutes an isomorphism. We must specify the requirements for the mapping itself and how it must interact with the vector space operations (addition and scalar multiplication).
General Strategy: To show two spaces are "isomorphic" , we need a map that is a perfect translation.
1. It must be a Linear Transformation (it respects the "math" of the space).
2. It must be a Bijection (every element in `bbb"V"` maps to exactly one unique element in `bbb"W"` , covering the whole space).
Specific Theory Used:
- Linearity: The map must satisfy `phi (u + v) = phi(u) + phi(v)` and `phi(cu) = c phi(u)`.
- Injectivity (One-to-One): If `phi(u) = phi(v)` , then `u = v`.
- Surjectivity (Onto): For every `w in bbb"W"` , there exists a `v in bbb"V"` such that `phi(v) = w`.
- Notation: When such a map exists , we write `bbb"V" cong bbb"W"`.
Solution: An isomorphism between a vector space `bbb"V"` and a vector space `bbb"W"` over a field `bbb"F"` is a bijective linear transformation `phi : bbb"V" rightarrow bbb"W"`. Formally , the mapping `phi` must satisfy these three properties for all `u , v in bbb"V"` and all `c in bbb"F"`:
1. Preservation of Addition: `phi(u + v) = phi(u) + phi(v)`.
2. Preservation of Scalar Multiplication: `phi(cu) = c phi(u)`.
3. Bijectivity: The map must be both injective (the kernel is only the zero vector) and surjective (the range of `phi` is all of `bbb"W"`). QED
Why this matters: If `bbb"V" cong bbb"W"` , any algebraic property true for `bbb"V"` is also true for `bbb"W"`. For example , if you find a basis in `bbb"V"` , its image under `phi` is automatically a basis for `bbb"W"`.
END
Question
Suppose `bbb"V"` is the vector space of all polynomials of degree at most 2
(e.g. `a x^2 + b x + c`) and `bbb"W"` is the space `bbb"F"^3` (ordered triples). Are these two spaces isomorphic
?
A) Yes , because we can map the coefficients directly to the three slots of the triple: `phi(a x^2 + b x + c) = (a \ , b \ , c)` .
B) No , because polynomials are functions and triples are just lists of numbers , they are fundamentally different types of objects.
