ElOperThry - 6 Elementary Linear Transform Theory Whether...

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Unformatted text preview: 6 Elementary Linear Transform Theory Whether they are spaces of “arrows in space”, functions or even matrices, vector spaces quickly become boring if we don’t do things with their elements — move them around, differentiate or integrate them, whatever. And often, what we do with these generalized vectors end up being linear operations. 6.1 Basic Definitions and Examples Let us suppose we have two vector spaces V and W (they could be the same vector space). Then we can certainly have a function L that transforms any vector v from V into a corresponding vector w = L ( v ) in W . This function is called a linear transformation (or linear transform or linear operator ) from V into W if and only if L (α 1 v 1 + α 2 v 2 ) = α 1 L ( v 1 ) + α 2 L ( v 2 ) whenever α 1 and α 2 are scalars, and v 1 and v 2 are vectors in V . This expression, of course, can be expanded to L parenleftBigg summationdisplay k α k v k parenrightBigg = summationdisplay k α k L ( v k ) for any linear combination ∑ k α k v k of vectors in V . (Remember, we are still insisting on linear combinations having only finitely many terms.) The domain of the operator is the vector space V , and the range is the set of all vectors in W given by L ( v ) where v ∈ V . On occasion, we might also call V the “input space”, and W the “target space”. This terminology is not standard, but is descriptive. Often, V and W will be the same. If this is the case, then we will simply refer to L as a linear transformation/transform/operator on V . It is also often true that W is not clearly stated. In such cases we can take W to be any vector space containing every L ( v ) for every v ∈ V . There is no requirement that every vector in W can be treated as L ( v ) for some v ∈ V . Here are a few examples of linear transforms on a traditional three-dimensional vector space V with { i , j , k } being a standard basis. In each case, the operator is defined by explaining what it does to an arbitrary vector v in V . Also given is a least one possible target space W . 10/2/2011 Elementary Linear Transform Theory Chapter & Page: 6–2 1. Any constant “magnification”, say, M 2 ( v ) = 2 v . Here, W = V . 2. Projection onto the k vector, −→ pr k ( v ) . Here, W is the one-dimensional space subspace of V consisting of all scalar multiples of k . (Actually, you can also view all of V as W .) 3. Projection onto the plane spanned by i and j , P (v 1 i + v 2 j + v 3 k ) = −→ pr { i , j } ( v ) = v 1 i + v 2 j . Here, W is the plane spanned by i and j . (Again, you can also view all of V as W .) 4. The cross product with some fixed vector a , say, a = 1 i + 2 j + 3 k , K a ( v ) = a × v ....
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ElOperThry - 6 Elementary Linear Transform Theory Whether...

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