5.2.1: Exercises 5.2

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In Exercises \(\PageIndex{1}\) – \(\PageIndex{5}\), a transformation \(T\) is given. Determine whether or not \(T\) is linear; if not, state why not.

Exercise \(\PageIndex{1}\)

\(T\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\right)=\left[\begin{array}{c}{x_{1}+x_{2}}\\{3x_{1}-x_{2}}\end{array}\right]\)

Answer: Yes

Exercise \(\PageIndex{2}\)

\(T\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\right)=\left[\begin{array}{c}{x_{1}+x_{2}^{2}}\\{x_{1}-x_{2}}\end{array}\right]\)

Answer: No; cannot have a squared term.

Exercise \(\PageIndex{3}\)

\(T\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\right)=\left[\begin{array}{c}{x_{1}+1}\\{x_{2}+1}\end{array}\right]\)

Answer: No; cannot add a constant.

Exercise \(\PageIndex{4}\)

\(T\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\right)=\left[\begin{array}{c}{1}\\{1}\end{array}\right]\)

Answer: No; cannot add a constant.

Exercise \(\PageIndex{5}\)

\(T\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\right)=\left[\begin{array}{c}{0}\\{0}\end{array}\right]\)

Answer: Yes

In Exercises \(\PageIndex{6}\) - \(\PageIndex{11}\), a linear transformation \(T\) is given. Find \([T]\).

Exercise \(\PageIndex{6}\)

\(T\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\right)=\left[\begin{array}{c}{x_{1}+x_{2}}\\{x_{1}-x_{2}}\end{array}\right]\)

Answer: \([T]=\left[\begin{array}{cc}{1}&{1}\\{1}&{-1}\end{array}\right]\)

Exercise \(\PageIndex{7}\)

\(T\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\right)=\left[\begin{array}{c}{x_{1}+2x_{2}}\\{3x_{1}-5x_{2}}\\{2x_{2}}\end{array}\right]\)

Answer: \([T]=\left[\begin{array}{cc}{1}&{2}\\{3}&{-5}\\{0}&{2}\end{array}\right]\)

Exercise \(\PageIndex{8}\)

\(T\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\\{x_{3}}\end{array}\right]\right)=\left[\begin{array}{c}{x_{1}+2x_{2}-3x_{3}}\\{0}\\{x_{1}+4x_{3}}\\{5x_{2}+x_{3}}\end{array}\right]\)

Answer: \([T]=\left[\begin{array}{ccc}{1}&{2}&{-3}\\{0}&{0}&{0}\\{1}&{0}&{4}\\{0}&{5}&{1}\end{array}\right]\)

Exercise \(\PageIndex{9}\)

\(T\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\\{x_{3}}\end{array}\right]\right)=\left[\begin{array}{c}{x_{1}+3x_{3}}\\{x_{1}-x_{3}}\\{x_{1}+x_{3}}\end{array}\right]\)

Answer: \([T]=\left[\begin{array}{ccc}{1}&{0}&{3}\\{1}&{0}&{-1}\\{1}&{0}&{1}\end{array}\right]\)

Exercise \(\PageIndex{10}\)

\(T\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\end{array}\right]\right)=\left[\begin{array}{c}{0}\\{0}\end{array}\right]\)

Answer: \([T]=\left[\begin{array}{cc}{0}&{0}\\{0}&{0}\end{array}\right]\)

Exercise \(\PageIndex{11}\)

\(T\left(\left[\begin{array}{c}{x_{1}}\\{x_{2}}\\{x_{3}}\\{x_{4}}\end{array}\right]\right)=[x_{1}+2x_{2}+3x_{3}+4x_{4}]\)

Answer: \([T]=[1\:2\:3\:4]\)