Problem 8
A linear transformation \(T\) is given. Find \([T]\). $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]\right)=\left[\begin{array}{c} x_{1}+2 x_{2}-3 x_{3} \\ 0 \\ x_{1}+4 x_{3} \\ 5 x_{2}+x_{3} \end{array}\right] $$
Problem 8
A list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane. (a) counterclockwise rotation by an angle of \(45^{\circ}\) (b) vertical stretch by a factor of \(1 / 2\)
Problem 9
A list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane. (a) clockwise rotation by an angle of \(90^{\circ}\) (b) horizontal reflection across the \(y\) axis (c) vertical shear by a factor of 1
Problem 9
In Exercises 9-12, a vector \(\vec{x}\) and a scalar \(a\) are given. Using Definition 32, compute the lengths of \(\vec{x}\) and \(a \vec{x},\) then compare these lengths. $$ \vec{x}=\left[\begin{array}{c} 1 \\ -2 \\ 5 \end{array}\right], a=2 $$
Problem 9
A linear transformation \(T\) is given. Find \([T]\). $$ T\left(\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]\right)=\left[\begin{array}{c} x_{1}+3 x_{3} \\ x_{1}-x_{3} \\ x_{1}+x_{3} \end{array}\right] $$
Problem 10
A list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane. (a) vertical reflection across the \(x\) axis (b) horizontal reflection across the \(y\) axis (c) diagonal reflection across the line \(y=x\)
Problem 11
In Exercises 11-14, two sets of transformations are given. Sketch the transformed unit square under each set of transformations. Are the transformations the same? Explain why/why not. (a) a horizontal reflection across the \(y\) axis, followed by a vertical reflection across the \(x\) axis, compared to (b) a counterclockise rotation of \(180^{\circ}\)
Problem 12
Two sets of transformations are given. Sketch the transformed unit square under each set of transformations. Are the transformations the same? Explain why/why not. (a) a horizontal stretch by a factor of 2 followed by a reflection across the line \(y=x,\) compared to (b) a vertical stretch by a factor of 2
Problem 12
A vector \(\vec{x}\) and a scalar \(a\) are given. Using Definition 32, compute the lengths of \(\vec{x}\) and \(a \vec{x},\) then compare these lengths. $$ \vec{x}=\left[\begin{array}{c} 1 \\ 2 \\ -2 \end{array}\right], a=3 $$
Problem 13
Two sets of transformations are given. Sketch the transformed unit square under each set of transformations. Are the transformations the same? Explain why/why not. (a) a horizontal stretch by a factor of \(1 / 2\) followed by a vertical stretch by a factor of \(3,\) compared to (b) the same operations but in opposite order
