Consider \(T:\R^2\rightarrow\R^2\) given by \(T\left[\left(\begin{array}{r}x\\y \end{array} \right)\right]=\left(\begin{array}{r} 2x+y\\4x-3y \end{array} \right)\text{.}\)
Show \(T\) is a linear operator.
Find a matrix \(A\) so that \(T\vec{x}=A\vec{x}\text{.}\)
Determine the range of \(T\text{.}\)
Determine the kernel of \(T\text{.}\)
Determine whether \(T\) is injective.
Determine whether \(T\) is surjective.
Consider \(T:\R^2\rightarrow\R^2\) given by \(T\left[\left(\begin{array}{r}x\\y \end{array} \right)\right]=\left(\begin{array}{c} 2x-y\\-6x+3y\end{array}\right)\text{.}\)
Show \(T\) is a linear operator.
Find a matrix \(A\) so that \(T\vec{x}=A\vec{x}\text{.}\)
Determine the range of \(T\text{.}\)
Determine the kernel of \(T\text{.}\)
Determine whether \(T\) is injective.
Determine whether \(T\) is surjective.
Consider \(T:\R^2\rightarrow\R^3\) given by \(T\left[\left(\begin{array}{r}x\\y\end{array} \right)\right]=\left(\begin{array}{c} x\\x+4z\\y-z\end{array}\right)\text{.}\)
Show \(T\) is a linear transformation.
Find a matrix \(A\) so that \(T\vec{x}=A\vec{x}\text{.}\)
Determine the range of \(T\text{.}\)
Determine the kernel of \(T\text{.}\)
Determine whether \(T\) is injective.
Determine whether \(T\) is surjective.
Consider the map \(T:\R^3 \rightarrow \R^2\) given by \(T\left[\left(\begin{array}{c} x\\y\\z \end{array} \right)\right]=\left(\begin{array}{c} 2x+7z\\y-2z \end{array} \right)\text{.}\)
Show \(T\) is a linear transformation.
Find a matrix \(A\) so that \(T\vec{x}=A\vec{x}\text{.}\)
Determine the range of \(T\text{.}\)
Determine the kernel of \(T\text{.}\)
Determine whether \(T\) is injective.
Determine whether \(T\) is surjective.
Find a matrix \(A\) that rotates each vector in \(\R^2\) counterclockwise about the origin by \(\pi/6\) radians.
Find a matrix \(A\) that reflects a vector in \(\R^2\) about the line \(y=-x\) by determining the mapping for the canonical basis vectors \(\wh{e}_1\) and \(\wh{e}_2\text{.}\)
A linear transformation \(T:\R^2 \rightarrow \R^2\) maps \(T\left[\left(\begin{array}{r} 3\\2 \end{array} \right)\right]=\left(\begin{array}{r} -1\\2 \end{array} \right)\text{,}\) and \(T\left[\left(\begin{array}{r} 7\\5 \end{array} \right)\right]=\left(\begin{array}{r} 2\\3 \end{array} \right)\text{.}\) Find a matrix \(A\) so that \(T\vec{x}=A\vec{x}\text{.}\)
Charlie and Eddy agree on a deal where Charlie will give Eddy one-half of his money and Eddy will give Charlie one-third of his money, simultaneously. Let \(x\) be the amount of money Charlie has and \(y\) the amount Eddy has before the exchange and let \(\vec{v}=\left(\begin{array}{r} x\\y \end{array} \right)\text{.}\) Let \(T(\vec{v})\) be the amount of money each man has after the exchange.
Find a matrix \(A\) so that \(T(\vec{v})=A\vec{v}\)
Is it possible for there to be a starting amount of money, \(x\) for Charlie and \(y\) for Eddy (other than $\(0\) each), so that \(T\left[\left(\begin{array}{r}x\\y \end{array} \right)\right]=\left(\begin{array}{r}x\\y \end{array} \right)?\) If so, find such an \(x\) and \(y\text{.}\) Is there more than one way for this to happen? Explain.
