Hi! I'm in need of some help regarding an assigment. I need to understand the concepts in the problem, and they are not explained in the literature we are using. This is the problem:(adsbygoogle = window.adsbygoogle || []).push({});

Given a linear transformation

[tex]\mathcal{A} = \mathbf{R}^n\rightarrow\mathbf{R}^m[/tex]

show that it can be divided/separated as [tex]\mathcal{A}=i\circ\mathcal{B}\circ{p}[/tex], where [tex]p[/tex] is the projection on the complement to the kernel, [tex]\lbrace \vec{v}; \mathcal{A}\vec{v} = \vec{0}\rbrace[/tex] (two elements in the complement are considered identical if they differ with one element in the kernel, ie we take the ratio???), the transformation [tex]\mathcal{B}[/tex] is an invertible transformation ([tex](\mathcal{B}\vec{v})^{-1} = \vec{v}[/tex]?), from the complement to the kernel to the image under [tex]\mathcal{A}[/tex], and [tex]i[/tex] is the inclusion of the image in [tex]R^m[/tex].

Ok, so the kernel is the set of vectors with the property [tex]\mathcal{A}\vec{v} = \vec{0}[/tex]. The complement should be [tex]\lbrace \vec{v}; \vec{v}\in\mathbf{R}^n \rbrace\backslash\lbrace\vec{v};\mathcal{A}\vec{v}=\vec{0}\rbrace[/tex]. Right? What about the projection on this set? How do you project a vector on a set? On every individual element, in which case the result is a set?

And the what is the inclusion [tex]i[/tex]? How should I approach this problem?

Thanks on advance,

Nille

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# Transformation assigment

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