Consider the operation of multiplying a vector in [itex]ℝ^{n}[/itex] by an [itex]m \times n [/itex] matrix A. This can be viewed as a linear transformation from [itex]ℝ^{n}[/itex] to [itex]ℝ^{m}[/itex]. Since matrices under matrix addition and multiplication by a scalar form a vector space, we can define a "vector space of linear transformations" from [itex]ℝ^{n}[/itex] to [itex]ℝ^{m}[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

My question is whether this connection between linear transformations and transformation matrices exists for linear mappings to and from arbitrary vector spaces. So given some general n-dimensional vector space U and m-dimensional vector space W, can every linear mapping from U to W be viewed as multiplication by a [itex]m \times n [/itex] transformation matrix ?

Or is there a linear transformation which cannot be viewed as multiplication by a transformation matrix?

BiP

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# The vector space of linear transformations

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