Problem 1
In Exercises 1-5, a transformation \(T\) is given. Determine whether or not \(T\) is linear; if not, state why not. $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{c} x_{1}+x_{2} \\ 3 x_{1}-x_{2} \end{array}\right] $$
Problem 2
A transformation T is given. Determine whether or not T is linear; if not, state why not. $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{l} x_{1}+x_{2}^{2} \\ x_{1}-x_{2} \end{array}\right] $$
Problem 3
A transformation T is given. Determine whether or not T is linear; if not, state why not. $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{l} x_{1}+1 \\ x_{2}+1 \end{array}\right] $$
Problem 4
A transformation T is given. Determine whether or not T is linear; if not, state why not. $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{l} 1 \\ 1 \end{array}\right] $$
Problem 5
In Exercises 5-10, a list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane. (a) vertical shear by a factor of 2 (b) horizontal shear by a factor of 2
Problem 5
A transformation T is given. Determine whether or not T is linear; if not, state why not. $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{l} 0 \\ 0 \end{array}\right] $$
Problem 6
In Exercises 6-11, a linear transformation \(T\) is given. Find \([T]\). $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{l} x_{1}+x_{2} \\ x_{1}-x_{2} \end{array}\right] $$
Problem 6
A list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane. (a) horizontal shear by a factor of 2 (b) vertical shear by a factor of 2
Problem 7
A list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane. (a) horizontal stretch by a factor of 3 (b) reflection across the line \(y=x\)
Problem 7
A linear transformation \(T\) is given. Find \([T]\). $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]\right)=\left[\begin{array}{c} x_{1}+2 x_{2} \\ 3 x_{1}-5 x_{2} \\ 2 x_{2} \end{array}\right] $$
