1. For a map f : V ?? W between vector spaces V and W to...

1. For a map f : V ?? W between vector spaces V and W to be a linear map it must preserve the structure of V . What must one verify to verify whether or not a map is linear?

2. For a map f : V ?? W between vector spaces to be an isomorphism it must be a linear map and also have two further properties. What are those two properties? As well as giving the names of the properties, explain what the names mean.

3.Every linear transformation is an isomorphism, but the isomorphism f : x y ?? x y is not a linear transformation. Why

Expert Solution

For a map f : V? W between vector spaces V and W to be a linear map, it must preserve the structure of V. We must verify that f preserves vector addition and also scalar multiplication i.e. if u and v are arbitrary vectors in V and if k is an arbitrary scalar, then f is a linear map if ana only if f(u+v) = f(u)+f(v) and f(ku) = kf(u).
For a map f : V? W between vector spaces to be an isomorphism it must be a linear map and also have two further properties.

An isomorphism is both injective or one-to-one and surjective or onto mapping. The mapping f : V? W is injective or one-to-one if f(u) = f(v). The mapping f : V ? W is surjective or onto if every w? W has a pre- image under f in V.

3. The statement is incorrect. Every linear transformation is an isomorphism, and every isomorphism f between 2 vector spaces is an invertible linear transformation. The notations given are not clear. Please elaborate.

milcah answered 1 year ago

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