Isomorphism of Vector Spaces Definition A linear transformation

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Isomorphism of Vector Spaces • Definition: A linear transformation T: V W is said

Isomorphism of Vector Spaces • Definition: A linear transformation T: V W is said to be an isomorphism if it is injective and surjective. • Proposition 27: Let V and W be finite-dimensional spaces. a) An isomorphism T: V W takes any arbitrary basis of V to a basis of W. b) Conversely, if a linear transformation T: V W takes some basis of V to a basis of W, then it is an isomorphism. • Proposition 28: Two finite-dimensional vector spaces V and W (over the same field F) are isomorphic if and only if dim V = dim W. • Proofs of Propositions 27 and 28 left as an exercise. • Remark: In particular, it follows that every vector space V of dimension n over R is isomorphic to Rn. As a consequence, we can certainly exploit our familiarity with Rn in proving results about finite-dimensional spaces.

A Very Important Linear Transformation - 1 • • Left Multiplication by a Matrix:

A Very Important Linear Transformation - 1 • • Left Multiplication by a Matrix: Let A be a fixed m n matrix. Then the function TA: Rn Rm defined by TA(x) = Ax is a linear transformation. Remark: This is very easy to see. You would have seen multiple examples while solving the problems related to linear transformations. In general, we do not need to distinguish between the matrix and the linear transformation TA.

A Very Important Linear Transformation - 2 • We now consider the reverse problem:

A Very Important Linear Transformation - 2 • We now consider the reverse problem: Suppose V and W are finite-dimensional vector spaces over the field F, and suppose T is a linear transformation T: V W. We will try to associate a matrix with this linear transformation.

Coordinate Systems • Observation 1: Given a basis for a finite-dimensional vector space V,

Coordinate Systems • Observation 1: Given a basis for a finite-dimensional vector space V, we recall that a vector can be expressed in one and only one way as a linear combination of the basis vectors. Therefore we make the following definitions: • Definition : An ordered basis for a finite-dimensional space V is a finite sequence of vectors which is linearly independent and spans V. In other words, an ordered basis is a basis with the vectors taken in a specified fixed order. • Given an ordered basis B = {u 1, u 2, …. , un} we can express any vector uniquely in the form u = x 1 u 1 + x 2 u 2 + …. . + xn un. The scalars xi are called the coordinates of u relative to the (ordered) basis B.