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Extension Theorem

Extension Theorem

Jul 25, 20251 min read

linear-algebra

Extension Theorem

Let V be a vector space and B be the set of basis vectors of V. Let W be a vector spaces and f:V→W be a linear mapping.

xf(x)​=x1​⋅b1​+…+xn​⋅bn​⟹=x1​⋅f(b1​)+…+xn​⋅f(bn​)​

where f(bi​) is called the image of the basis vector bi​.

Conversely, this formula always defines a linear mapping f:V→W for any choice of f(bj​)∈W. Thus, it is enough to know the images of the basis vectors to characterise a linear mapping f:V→W.


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