Let \(T: \IR^n \rightarrow \IR^m\) be a linear map with standard matrix \(A\text{.}\) Sort the following items into three groups of statements: a group that means \(T\) is injective, a group that means \(T\) is surjective, and a group that means \(T\) is bijective.

1. \(A\vec x=\vec b\) has a solution for all \(\vec b\in\IR^m\)

2. \(A\vec x=\vec b\) has a unique solution for all \(\vec b\in\IR^m\)

3. \(A\vec x=\vec 0\) has a unique solution.

4. The columns of \(A\) span \(\IR^m\)

5. The columns of \(A\) are linearly independent

6. The columns of \(A\) are a basis of \(\IR^m\)

7. Every column of \(\RREF(A)\) has a pivot

8. Every row of \(\RREF(A)\) has a pivot

9. \(m=n\) and \(\RREF(A)=I\)

Let \(T: \IR^3 \rightarrow \IR^3\) be the linear transformation given by the standard matrix \(A=\left[\begin{array}{ccc} 2 & -1 & 0 \\ 2 & 1 & 4 \\ 1 & 1 & 3 \end{array}\right]\text{.}\)

Write an augmented matrix representing the system of equations given by \(T(\vec x)=\vec{0}\text{,}\) that is, \(A\vec x=\left[\begin{array}{c}0 \\ 0 \\ 0 \end{array}\right]\text{.}\) Then solve \(T(\vec x)=\vec{0}\) to find the kernel of \(T\text{.}\)

Let \(T: \IR^n \rightarrow \IR^n\) be a linear map with standard matrix \(A\text{.}\)

• If \(T\) is a bijection and \(\vec b\) is any \(\IR^n\) vector, then \(T(\vec x)=A\vec x=\vec b\) has a unique solution.

• So we may define an inverse map \(T^{-1} : \IR^n \rightarrow \IR^n\) by setting \(T^{-1}(\vec b)\) to be this unique solution.

• Let \(A^{-1}\) be the standard matrix for \(T^{-1}\text{.}\) We call \(A^{-1}\) the inverse matrix of \(A\text{,}\) so we also say that \(A\) is invertible.

Let \(T: \IR^3 \rightarrow \IR^3\) be the linear transformation given by the standard matrix \(A=\left[\begin{array}{ccc} 2 & -1 & -6 \\ 2 & 1 & 3 \\ 1 & 1 & 4 \end{array}\right]\text{.}\)

Part 1.

Write an augmented matrix representing the system of equations given by \(T(\vec x)=\vec{e}_1\text{,}\) that is, \(A\vec x=\left[\begin{array}{c}1 \\ 0 \\ 0 \end{array}\right]\text{.}\) Then solve \(T(\vec x)=\vec{e}_1\) to find \(T^{-1}(\vec{e}_1)\text{.}\)

Let \(T: \IR^3 \rightarrow \IR^3\) be the linear transformation given by the standard matrix \(A=\left[\begin{array}{ccc} 2 & -1 & -6 \\ 2 & 1 & 3 \\ 1 & 1 & 4 \end{array}\right]\text{.}\)

Part 2.

Solve \(T(\vec x)=\vec{e}_2\) to find \(T^{-1}(\vec{e}_2)\text{.}\)

Let \(T: \IR^3 \rightarrow \IR^3\) be the linear transformation given by the standard matrix \(A=\left[\begin{array}{ccc} 2 & -1 & -6 \\ 2 & 1 & 3 \\ 1 & 1 & 4 \end{array}\right]\text{.}\)

Part 3.

Solve \(T(\vec x)=\vec{e}_3\) to find \(T^{-1}(\vec{e}_3)\text{.}\)

Let \(T: \IR^3 \rightarrow \IR^3\) be the linear transformation given by the standard matrix \(A=\left[\begin{array}{ccc} 2 & -1 & -6 \\ 2 & 1 & 3 \\ 1 & 1 & 4 \end{array}\right]\text{.}\)

Part 4.

Write \(A^{-1}\text{,}\) the standard matrix for \(T^{-1}\text{.}\)

We could have solved these three systems simultaneously by row reducing the matrix \([A\,|\,I]\) at once.

Find the inverse \(A^{-1}\) of the matrix \(A=\left[\begin{array}{cc} 1 & 3 \\ 0 & -2 \end{array}\right]\) by row-reducing \([A\,|\,I]\text{.}\)

Is the matrix \(\left[\begin{array}{ccc} 2 & 3 & 1 \\ -1 & -4 & 2 \\ 0 & -5 & 5 \end{array}\right]\) invertible? Give a reason for your answer.

An \(n\times n\) matrix \(A\) is invertible if and only if \(\RREF(A) = I_n\text{.}\)

Let \(T:\IR^2\to\IR^2\) be the bijective linear map defined by \(T\left(\left[\begin{array}{c}x\\y\end{array}\right]\right)=\left[\begin{array}{c} 2x -3y \\ -3x + 5y\end{array}\right]\text{,}\) with the inverse map \(T^{-1}\left(\left[\begin{array}{c}x\\y\end{array}\right]\right)=\left[\begin{array}{c} 5x+ 3y \\ 3x + 2y\end{array}\right]\text{.}\)

Part 1.

Compute \((T^{-1}\circ T)\left(\left[\begin{array}{c}-2\\1\end{array}\right]\right)\text{.}\)

Let \(T:\IR^2\to\IR^2\) be the bijective linear map defined by \(T\left(\left[\begin{array}{c}x\\y\end{array}\right]\right)=\left[\begin{array}{c} 2x -3y \\ -3x + 5y\end{array}\right]\text{,}\) with the inverse map \(T^{-1}\left(\left[\begin{array}{c}x\\y\end{array}\right]\right)=\left[\begin{array}{c} 5x+ 3y \\ 3x + 2y\end{array}\right]\text{.}\)

Part 2.

If \(A\) is the standard matrix for \(T\) and \(A^{-1}\) is the standard matrix for \(T^{-1}\text{,}\) find the \(2\times 2\) matrix

\(T^{-1}\circ T=T\circ T^{-1}\) is the identity map for any bijective linear transformation \(T\text{.}\) Therefore \(A^{-1}A=AA^{-1}\) equals the identity matrix \(I\) for any invertible matrix \(A\text{.}\)