Linear Transformations
Part 5: Summary
 Give the definition of a linear transformation.
 Explain why any linear transformation from R^{n} to R^{m} is completely determined by where it sends the standard unit vectors e_{1}, ... , e_{n} of R^{n}.
Be sure to explain how you can find T(x) for any vector x so long as you know T(e_{1}), ..., T(e_{n}).
 Justify the following statement:
Any linear transformation T from R^{n} to R^{m} is a matrix transformation, meaning that there is an m x n matrix A such that T(x) = Ax for all vectors x in R^{n}.
You should explain how to find the matrix A.
 Describe the geometric transformation defined by each of the matrices
Challenge question: Explain how each transformation could be seen as a composition of two other transformations.

Find the matrix of a linear transformation that takes the unit square to a
trapezoid with vertices (0,0), (1.5,1), (0.5,1), and (1,0).
 Clearly and concisely explain the relationships between the linear (in)dependence of rows/columns of a matrix and the injectivity/surjectivity of the linear transformation defined by the matrix.
 Does every matrix define a linear transformation? Explain your answer.
 Is every linear transformation defined by a matrix? Explain your answer.
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