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Exercises 3.9 Practice Exam on Linear Transformations and Projections

Practice Exercises:

Here are some exercises from Hefferon's text which you should try while studying for the exam, the answers to all of these can be found in the Answers to Exercises supplement for Hefferon's text. You do not need to hand in these exercises.

Additionally, be sure you can define the following: Linear Transformations, Matrix of a Transformation, One-to-One, Onto, Kernel, Zero Divisor, Transpose, Inverse Matrix, Null Space, Projection, Orthogonal Vectors, Orthogonal Complement, Projection of a vector onto a subspace, Orthonormal Basis, Change of Basis Matrix (also called change of coordinate matrix)

Practice Exam:

In addition to the practice problems above below are the questions for your practice exam, these must be turned in when you come in for the exam. The practice exam counts for 5% of your exam grade and can earn you back 15% of any points you loose on the in-class portion of the exam.


Define linear transformation.


Define inverse matrix.


Define one-to-one function.


Define kernel of a transformation.


Define orthogonal complement.


Find the inverse of the given matrix: (compare to p.252)

\begin{equation*} A=\left[ \begin{array}{rrr} 1 \amp 2 \amp 4\\ -1 \amp 0 \amp 2\\ 0 \amp 3 \amp 10 \end{array} \right] \end{equation*}

Find the kernel (i.e. null space) for the transformation defined by multiplication by: (compare to p.200)

\begin{equation*} A=\left[ \begin{array}{rrr} 3 \amp 2 \amp 5\\ -1 \amp 0 \amp 2 \end{array} \right] \end{equation*}

Is the transformation defined by the given matrix one-to-one? Is it onto? (compare to p.272)

\begin{equation*} B=\left[ \begin{array}{rr} 1 \amp 2 \\ -3 \amp -5\\ 4 \amp 7\\ \end{array} \right] \end{equation*}

Find the domain, codomain, and range for the transformation defined by the matrix. (compare to p.220)

\begin{equation*} C=\left[ \begin{array}{rrrr} 1 \amp 0 \amp 3 \amp 0\\ -3 \amp 1\amp -5\amp 1\\ 2 \amp 1\amp 2\amp 2\\ \end{array} \right] \end{equation*}

Find the transformation, \(Rep_{\mathcal{E}_3,\mathcal{B}}\text{,}\) from the standard basis, \(\mathcal{E}_3\text{,}\) to the basis \(\mathcal{B}\text{.}\) (compare to p.200)

\begin{equation*} \mathcal{E}_3= \left\{ \left( \begin{array}{r} 1 \\ 0 \\0 \end{array} \right), \left( \begin{array}{r} 0 \\ 1 \\0 \end{array} \right), \left( \begin{array}{r} 0 \\ 0 \\1 \end{array} \right) \right\},\ \mathcal{B}= \left\{ \left( \begin{array}{r} 1 \\ 0 \\-1 \end{array} \right), \left( \begin{array}{r} 0 \\ 1 \\1 \end{array} \right), \left( \begin{array}{r} 1 \\ 1 \\2 \end{array} \right) \right\} \end{equation*}

Use your answer from Exercise 3.9.10 to find the transformation from \(\mathcal{C}\) to \(\mathcal{B}\text{,}\) \(Rep_{\mathcal{C}, \mathcal{B}}\text{.}\) (compare to p.257)

\begin{equation*} \mathcal{C}= \left\{ \left( \begin{array}{r} 1 \\ 0 \\ 0 \end{array} \right), \left( \begin{array}{r} 1 \\ 1 \\0 \end{array} \right), \left( \begin{array}{r} 1 \\ 1 \\1 \end{array} \right) \right\} \end{equation*}

Find the orthogonal compliment of the subspace defined by the set of vectors: (compare to p.283)

\begin{equation*} \mathcal{D}= \left\{ \left( \begin{array}{r} 1 \\ 0 \\ 0 \\ 1 \end{array} \right), \left( \begin{array}{r} 0 \\ 0 \\ 2 \\ -1 \end{array} \right) \right\} \end{equation*}

Use the Gram-Schmidt process to create and orthogonal basis to replace \(\mathcal{D}\text{.}\) (compare to p.275)

\begin{equation*} \mathcal{D}= \left\{ \left( \begin{array}{r} 1 \\ 1 \\1 \end{array} \right), \left( \begin{array}{r} 0 \\ 2 \\0 \end{array} \right), \left( \begin{array}{r} 0 \\ 1 \\1 \end{array} \right) \right\} \end{equation*}

Use linear algebra to find the line of best fit for the points \((1,5)\text{,}\) \((3,3)\text{,}\) and \((4,0)\text{.}\) (compare to p.290 and p.283)

Figure 3.9.1. Least Squares Problem