Linear Transformations — Practice Problems and Worked Solutions

17 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Evaluate the linear transformation.

   $$T(x)=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]x,x=\left[\begin{array}{c}2\\ 1\end{array}\right]$$

   Answer: $\left[\begin{array}{c}4\\ 1\end{array}\right]$

   Step-by-step solution

   1. $$T(x)=Ax$$
   2. $$T(x)=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]\left[\begin{array}{c}2\\ 1\end{array}\right]=\left[\begin{array}{c}4\\ 1\end{array}\right]$$
2. 2.

   Evaluate the linear transformation.

   $$T(x)=\left[\begin{array}{cc}2& -1\\ 3& 0\end{array}\right]x,x=\left[\begin{array}{c}1\\ 4\end{array}\right]$$

   Answer: $\left[\begin{array}{c}-2\\ 3\end{array}\right]$

   Step-by-step solution

   1. $$T(x)=Ax$$
   2. $$T(x)=\left[\begin{array}{cc}2& -1\\ 3& 0\end{array}\right]\left[\begin{array}{c}1\\ 4\end{array}\right]=\left[\begin{array}{c}-2\\ 3\end{array}\right]$$
3. 3.

   Evaluate the linear transformation.

   $$T(x)=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& -1\end{array}\right]x,x=\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right]$$

   Answer: $\left[\begin{array}{c}4\\ 2\end{array}\right]$

   Step-by-step solution

   1. $$T(x)=Ax$$
   2. $$T(x)=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& -1\end{array}\right]\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right]=\left[\begin{array}{c}4\\ 2\end{array}\right]$$
4. 4.

   Evaluate the linear transformation.

   $$T(x)=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]x,x=\left[\begin{array}{c}3\\ 2\end{array}\right]$$

   Answer: $\left[\begin{array}{c}-2\\ 3\end{array}\right]$

   Step-by-step solution

   1. $$T(x)=Ax$$
   2. $$T(x)=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{c}3\\ 2\end{array}\right]=\left[\begin{array}{c}-2\\ 3\end{array}\right]$$
5. 5.

   Evaluate the linear transformation.

   $$T(x)=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]x,x=\left[\begin{array}{c}-1\\ 3\end{array}\right]$$

   Answer: $\left[\begin{array}{c}1\\ 5\end{array}\right]$

   Step-by-step solution

   1. $$T(x)=Ax$$
   2. $$T(x)=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]\left[\begin{array}{c}-1\\ 3\end{array}\right]=\left[\begin{array}{c}1\\ 5\end{array}\right]$$
6. 6.

   Evaluate the linear transformation.

   $$T(x)=\left[\begin{array}{ccc}1& 2& 0\\ -1& 0& 3\end{array}\right]x,x=\left[\begin{array}{c}2\\ -1\\ 4\end{array}\right]$$

   Answer: $\left[\begin{array}{c}0\\ 10\end{array}\right]$

   Step-by-step solution

   1. $$T(x)=Ax$$
   2. $$T(x)=\left[\begin{array}{ccc}1& 2& 0\\ -1& 0& 3\end{array}\right]\left[\begin{array}{c}2\\ -1\\ 4\end{array}\right]=\left[\begin{array}{c}0\\ 10\end{array}\right]$$
7. 7.

   Evaluate the linear transformation.

   $$T(x)=\left[\begin{array}{cc}1& 1\\ 0& 2\end{array}\right]x,x=\left[\begin{array}{c}3\\ 1\end{array}\right]$$

   Answer: $\left[\begin{array}{c}4\\ 2\end{array}\right]$

   Step-by-step solution

   1. $$T(x)=Ax$$
   2. $$T(x)=\left[\begin{array}{cc}1& 1\\ 0& 2\end{array}\right]\left[\begin{array}{c}3\\ 1\end{array}\right]=\left[\begin{array}{c}4\\ 2\end{array}\right]$$
8. 8.

   Evaluate the linear transformation.

   $$T(x)=\left[\begin{array}{cc}2& 0\\ 0& 3\end{array}\right]x,x=\left[\begin{array}{c}1\\ 2\end{array}\right]$$

   Answer: $\left[\begin{array}{c}2\\ 6\end{array}\right]$

   Step-by-step solution

   1. $$T(x)=Ax$$
   2. $$T(x)=\left[\begin{array}{cc}2& 0\\ 0& 3\end{array}\right]\left[\begin{array}{c}1\\ 2\end{array}\right]=\left[\begin{array}{c}2\\ 6\end{array}\right]$$
9. 9.

   Evaluate the linear transformation.

   $$T(x)=\left[\begin{array}{cc}3& -1\\ 1& 2\end{array}\right]x,x=\left[\begin{array}{c}2\\ 1\end{array}\right]$$

   Answer: $\left[\begin{array}{c}5\\ 4\end{array}\right]$

   Step-by-step solution

   1. $$T(x)=Ax$$
   2. $$T(x)=\left[\begin{array}{cc}3& -1\\ 1& 2\end{array}\right]\left[\begin{array}{c}2\\ 1\end{array}\right]=\left[\begin{array}{c}5\\ 4\end{array}\right]$$
10. 10.

    Evaluate the linear transformation.

    $$T(x)=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]x,x=\left[\begin{array}{c}4\\ 5\end{array}\right]$$

    Answer: $\left[\begin{array}{c}5\\ -4\end{array}\right]$

    Step-by-step solution

    1. $$T(x)=Ax$$
    2. $$T(x)=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\left[\begin{array}{c}4\\ 5\end{array}\right]=\left[\begin{array}{c}5\\ -4\end{array}\right]$$
11. 11.

    Evaluate the linear transformation.

    $$T(x)=\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]x,x=\left[\begin{array}{c}1\\ -2\end{array}\right]$$

    Answer: $\left[\begin{array}{c}-5\\ 0\end{array}\right]$

    Step-by-step solution

    1. $$T(x)=Ax$$
    2. $$T(x)=\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]\left[\begin{array}{c}1\\ -2\end{array}\right]=\left[\begin{array}{c}-5\\ 0\end{array}\right]$$
12. 12.

    Evaluate the linear transformation.

    $$T(x)=\left[\begin{array}{ccc}2& 1& 0\\ 1& 0& 3\end{array}\right]x,x=\left[\begin{array}{c}1\\ 2\\ 1\end{array}\right]$$

    Answer: $\left[\begin{array}{c}4\\ 4\end{array}\right]$

    Step-by-step solution

    1. $$T(x)=Ax$$
    2. $$T(x)=\left[\begin{array}{ccc}2& 1& 0\\ 1& 0& 3\end{array}\right]\left[\begin{array}{c}1\\ 2\\ 1\end{array}\right]=\left[\begin{array}{c}4\\ 4\end{array}\right]$$
13. 13.

    Evaluate the linear transformation.

    $$T(x)=\left[\begin{array}{cc}4& 0\\ 0& 4\end{array}\right]x,x=\left[\begin{array}{c}2\\ -3\end{array}\right]$$

    Answer: $\left[\begin{array}{c}8\\ -12\end{array}\right]$

    Step-by-step solution

    1. $$T(x)=Ax$$
    2. $$T(x)=\left[\begin{array}{cc}4& 0\\ 0& 4\end{array}\right]\left[\begin{array}{c}2\\ -3\end{array}\right]=\left[\begin{array}{c}8\\ -12\end{array}\right]$$
14. 14.

    Evaluate the linear transformation.

    $$T(x)=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]x,x=\left[\begin{array}{c}5\\ 1\end{array}\right]$$

    Answer: $\left[\begin{array}{c}7\\ 19\end{array}\right]$

    Step-by-step solution

    1. $$T(x)=Ax$$
    2. $$T(x)=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{c}5\\ 1\end{array}\right]=\left[\begin{array}{c}7\\ 19\end{array}\right]$$
15. 15.

    Evaluate the linear transformation.

    $$T(x)=\left[\begin{array}{ccc}0& 2& 1\\ 1& 1& 0\end{array}\right]x,x=\left[\begin{array}{c}3\\ 1\\ 2\end{array}\right]$$

    Answer: $\left[\begin{array}{c}4\\ 4\end{array}\right]$

    Step-by-step solution

    1. $$T(x)=Ax$$
    2. $$T(x)=\left[\begin{array}{ccc}0& 2& 1\\ 1& 1& 0\end{array}\right]\left[\begin{array}{c}3\\ 1\\ 2\end{array}\right]=\left[\begin{array}{c}4\\ 4\end{array}\right]$$
16. 16.

    Evaluate the linear transformation.

    $$T(x)=\left[\begin{array}{cc}5& -2\\ 1& 1\end{array}\right]x,x=\left[\begin{array}{c}2\\ 3\end{array}\right]$$

    Answer: $\left[\begin{array}{c}4\\ 5\end{array}\right]$

    Step-by-step solution

    1. $$T(x)=Ax$$
    2. $$T(x)=\left[\begin{array}{cc}5& -2\\ 1& 1\end{array}\right]\left[\begin{array}{c}2\\ 3\end{array}\right]=\left[\begin{array}{c}4\\ 5\end{array}\right]$$
17. 17.

    Evaluate the linear transformation.

    $$T(x)=\left[\begin{array}{cc}1& 0\\ 2& 3\end{array}\right]x,x=\left[\begin{array}{c}4\\ 2\end{array}\right]$$

    Answer: $\left[\begin{array}{c}4\\ 14\end{array}\right]$

    Step-by-step solution

    1. $$T(x)=Ax$$
    2. $$T(x)=\left[\begin{array}{cc}1& 0\\ 2& 3\end{array}\right]\left[\begin{array}{c}4\\ 2\end{array}\right]=\left[\begin{array}{c}4\\ 14\end{array}\right]$$

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