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Linear Transformations

Part 5: Summary

  1. Give the definition of a linear transformation.

  2. Explain why any linear transformation from Rn to Rm is completely determined by where it sends the standard unit vectors e1, ... , en of Rn. Be sure to explain how you can find T(x) for any vector x so long as you know T(e1), ..., T(en).

  3. Justify the following statement:
    Any linear transformation T from Rn to Rm is a matrix transformation, meaning that there is an m x n matrix A such that T(x) = Ax for all vectors x in Rn.
    You should explain how to find the matrix A.

  4. Describe the geometric transformation defined by each of the matrices

    , , and .

    Challenge question: Explain how each transformation could be seen as a composition of two other transformations.

  5. 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).

  6. 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.

  7. Does every matrix define a linear transformation? Explain your answer.

  8. Is every linear transformation defined by a matrix? Explain your answer.

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