- #1

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Here's my line of reasoning:

If the set of vectors [tex]V = \left\{ v_1,v_2\right\}[/tex] is a basis for the 2-dimensional vector space [tex]X[/tex] and [tex]x \in X[/tex], then let

[tex] \left(x\right)_V = \left( c_1,c_2\right)[/tex]

denote the component vector of [tex]x[/tex] with respect to the basis [tex]V[/tex]. Now, let [tex]E[/tex] be the standard basis for [tex]X[/tex]; i.e.,

[tex]E = \left\{ \left(1,0\right),\left(0,1\right)\right\}[/tex]. Suppose

[tex]\left(v_1\right)_E = \left(2,1\right),[/tex]

and

[tex] \left(v_2\right)_E = \left(0,1\right)[/tex].

If [tex]\left(x\right)_E = \left(2,3\right)[/tex], then

[tex] \left(x\right)_V = \left(1,2\right)[/tex].

However, if we use the standard euclidean norm, the norm of vector [tex]\left(x\right)_V[/tex] is [tex]\sqrt{5}[/tex], whereas the norm of [tex]\left(x\right)_E[/tex] is [tex]\sqrt{13}[/tex].

Is this a correct analysis? It seems correct, since the euclidean norm depends on the components of the vector, and the components depend on the choice of basis...but something seems fishy.

Thanks!