Question

In each case, show that the linear transformation \(T\) satisfies \(T^{2}=T\). a. \(T: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4} ; T(x, y, z, w)=(x, 0, z, 0)\) b. \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} ; T(x, y)=(x+y, 0)\) c. \(T: \mathbf{P}_{2} \rightarrow \mathbf{P}_{2}\) \(\quad T\left(a+b x+c x^{2}\right)=(a+b-c)+c x+c x^{2}\) d. \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}\) \(\quad T\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=\frac{1}{2}\left[\begin{array}{ll}a+c & b+d \\ a+c & b+d\end{array}\right]\)

Short answer

In each case, \(T^2 = T\).

Step-by-step solution

Define T^2 for Part (a)

For the transformation \(T: \mathbb{R}^4 \rightarrow \mathbb{R}^4; T(x, y, z, w) = (x, 0, z, 0)\), we apply \(T\) twice to get \(T(T(x, y, z, w))\). Compute the first transformation: \(T(x, y, z, w) = (x, 0, z, 0)\). Applying \(T\) again yields \(T(x, 0, z, 0) = (x, 0, z, 0)\). Since applying \(T\) twice results in the same transformation, \(T^2 = T\).

Define T^2 for Part (b)

Here, the transformation is \(T: \mathbb{R}^2 \rightarrow \mathbb{R}^2; T(x, y) = (x+y, 0)\). Applying \(T\) twice gives us \(T(T(x, y))\). First, we compute \(T(x, y) = (x+y, 0)\). Applying \(T\) again to this result: \(T(x+y, 0) = (x+y+0, 0) = (x+y, 0)\). As the transformation yields the same result when applied twice, \(T^2 = T\).

Define T^2 for Part (c)

Consider the transformation \(T: \mathbf{P}_2 \rightarrow \mathbf{P}_2\) given by \(T(a + bx + cx^2) = (a+b-c) + cx + cx^2\). Applying \(T\) once gives us: \(T(a+b-c + cx + cx^2) = (a+b-c+c-c) + c\cdot x + c\cdot x^2 = (a+b-c) + cx + cx^2\). As \(T\) remains the same after another application, \(T^2 = T\).

Define T^2 for Part (d)

Here, the transformation is \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}\) defined by \(T\left[\begin{array}{ll}a & b \ c & d\end{array}\right] = \frac{1}{2}\left[\begin{array}{ll}a+c & b+d \ a+c & b+d\end{array}\right]\). Apply \(T\) once: get \(\frac{1}{2}\left[\begin{array}{ll}a+c & b+d \ a+c & b+d\end{array}\right]\). Applying \(T\) again yields the same matrix since each cell is the sum of corresponding rows and columns. Thus, \(T^2 = T\).

Key concepts

Linear Algebra

Linear Algebra is a fundamental area of mathematics. It explores vectors, vector spaces, and linear transformations among other concepts. A linear transformation is a mapping between two vector spaces that preserves operations of addition and scalar multiplication. This means if you have two vectors, say \(\mathbf{u} \) and \(\mathbf{v}\), and a scalar \(c\), a linear transformation \(T\) will satisfy the properties: \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \) and \( T(c \cdot \mathbf{u}) = c \cdot T(\mathbf{u}) \).

In the original exercise, you are examining transformations such as \(T: \mathbb{R}^4 \rightarrow \mathbb{R}^4\) or \(T: \mathbf{M}_{22} \rightarrow \mathbf{M}_{22}\). These mappings are specific examples of transformations acting on different vector spaces, illustrating the diverse applications of linear algebra solutions. Linear transformations are integral in solving equations that model real-world scenarios, making linear algebra relevant in engineering, physics, and computer science.

Transformation Matrices

A transformation matrix is a powerful tool in linear algebra. It allows us to express a linear transformation in terms of matrix multiplication. For example, for a transformation \(T\) that acts on elements of \(\mathbb{R}^n\), there exists a matrix \(A\) such that for any vector \(\mathbf{x}\), \( T(\mathbf{x}) = A\mathbf{x} \).

Transformation matrices simplify computations of linear transformations by reducing them to matrix multiplication. Given a transformation matrix \(A\), applying the transformation \(T\) multiple times corresponds to multiplying \(A\) by itself. This forms the basis for concepts like eigendecomposition in linear algebra, where matrices can be raised to various powers to analyze their behavior.

In the provided solutions, each case of \(T\) was analyzed to prove \(T^2 = T\). Essentially, this means that multiplying the matrix representation of \(T\) by itself returns the same matrix, reinforcing the unique property of certain matrices being projections.

Invariant Transformations

Invariant transformations are a fascinating aspect of transformations that remain unchanged under certain operations. In linear algebra, a transformation \(T\) is invariant if applying it more than once yields the same result: \(T^2 = T\). This property characterizes a specific category of transformations known as projections.

Projection transformations are particularly important in applications such as computer graphics and signal processing. They help in reducing dimensionality, such as projecting a 3D object onto a 2D plane.

In the exercise solutions, each transformation \(T\) was shown to be invariant (\(T^2 = T\)) by verifying that performing the transformation twice gives back the same result. This emphasizes the consistency and stability of such transformations under repeated applications, a desirable property in many analytical processes.