Question

Determine whether each of the following transformations \(T\) has an inverse and, if so, determine the action of \(T^{-1}\). a. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) \(\quad T(x, y, z)=(x+y, y+z, z+x)\) b. \(T: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) \(\quad T(x, y, z, t)=(x+y, y+z, z+t, t+x)\) c. \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}\) \(\quad T\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=\left[\begin{array}{cc}a-c & b-d \\ 2 a-c & 2 b-d\end{array}\right]\) d. \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}\) \(T\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=\left[\begin{array}{ll}a+2 c & b+2 d \\ 3 c-a & 3 d-b\end{array}\right]\) e. \(T: \mathbf{P}_{2} \rightarrow \mathbb{R}^{3} ; T\left(a+b x+c x^{2}\right)=(a-c, 2 b, a+c)\) f. \(T: \mathbf{P}_{2} \rightarrow \mathbb{R}^{3} ; T(p)=[p(0), p(1), p(-1)]\)

Short answer

Only b, c, and e have inverses; d and f do not have inverses.

Step-by-step solution

Determine the inverse of transformation a

To check if the function \( T(x, y, z) = (x+y, y+z, z+x) \) is invertible, represent it in terms of linear equations:\[\begin{align*} u &= x + y, \ v &= y + z, \ w &= z + x.\end{align*}\]We can express this system as three equations with three unknowns:\[\begin{align*} u &= x + y, \ v &= y + z, \ w &= z + x.\end{align*}\]To solve for \(x\), \(y\), and \(z\), notice that:- From \( u + v + w \) we get \( 2(x + y + z) = u + v + w \)... thus, expressing:\[ x = \frac{u - v + w}{2}, \ y = \frac{u + v - w}{2}, \ z = \frac{-u + v + w}{2}.\]So, \( T^{-1}(u, v, w) = \left( \frac{u - v + w}{2}, \frac{u + v - w}{2}, \frac{-u + v + w}{2} \right) \).

Key concepts

Invertibility of Functions

A function is considered invertible if every output uniquely maps back to a single input. To determine whether a transformation is invertible, we check if we can reverse the process. This involves ensuring that given any output, there is one and only one input that produces it.

For a linear transformation represented by the function \(T\), invertibility requires that \(T\) is a bijective mapping. This means the transformation must be both one-to-one (injective) and onto (surjective). In practical terms, this involves ensuring that the transformation matrix has full rank, which results in the determinant being non-zero.

In the original problem, to determine if the transformation \(T(x, y, z) = (x+y, y+z, z+x)\) is invertible, we set up equations and found expressions for \(x\), \(y\), and \(z\). The crucial step was ensuring there was a unique solution, indicating the presence of an inverse and thus confirming the invertibility of \(T\).

Matrices and Determinants

Matrices provide a structured way to represent linear transformations. Understanding the concept of invertibility relies heavily on the determinant of the transformation matrix. The determinant tells us about the scaling of vectors transformed by the matrix and whether such transformations can be reversed.

The determinant provides crucial information:

• If the determinant of the matrix is not zero, the transformation is invertible, and an inverse matrix exists.
• If it is zero, the matrix is singular, meaning the transformation compresses by some dimension, losing information in a way that can't be reversed.

In cases where rotation, reflection, or other complex mappings occur, analyzing the corresponding matrix's determinant offers clues about the behavior and reversibility of the transformation.

Consider a matrix transformation \(A\) from \(\mathbb{R}^3\) to \(\mathbb{R}^3\). If we were to calculate its determinant and find it was non-zero, we could confidently determine the matrix \(A\) as invertible, allowing us to find \(A^{-1}\) and solve the original equations for any set vector.

Polynomial Mappings

Polynomial mappings describe transformations where the relationship between input and output values is given by a polynomial. These mappings are often more complex due to the nature of polynomial functions which can span multiple dimensions.

In the given exercise, transformations such as \(T(p) = [p(0), p(1), p(-1)]\) express polynomial mappings, where \(p\) is a polynomial and the outputs are real numbers related to the values of the polynomial at specific points. To determine the invertibility of such a mapping, we observe if each polynomial input uniquely and consistently transforms into the outputs.

Polynomial mappings often require checking the independence of the functions defined by the polynomial to ensure no two distinct polynomial functions yield the same output vector. In simpler terms, we need to establish a clear one-to-one correspondence between any polynomial and its output vector, ensuring it fully covers all potential outputs.