Question

Show that each of the following functions is a linear transformation. a. \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} ; T(x, y)=(x,-y)\) (reflection in the \(x\) axis) b. \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} ; T(x, y, z)=(x, y,-z)\) (reflection in the \(x\) - \(y\) plane) c. \(T: \mathbb{C} \rightarrow \mathbb{C} ; T(z)=\bar{z}\) (conjugation) d. \(T: \mathbf{M}_{m n} \rightarrow \mathbf{M}_{k l} ; T(A)=P A Q, P\) a \(k \times m\) matrix, \(Q\) an \(n \times l\) matrix, both fixed e. \(T: \mathbf{M}_{n n} \rightarrow \mathbf{M}_{n n} ; T(A)=A^{T}+A\) f. \(T: \mathbf{P}_{n} \rightarrow \mathbb{R} ; T[p(x)]=p(0)\) g. \(T: \mathbf{P}_{n} \rightarrow \mathbb{R} ; T\left(r_{0}+r_{1} x+\cdots+r_{n} x^{n}\right)=r_{n}\) h. \(T: \mathbb{R}^{n} \rightarrow \mathbb{R} ; T(\mathbf{x})=\mathbf{x} \cdot \mathbf{z}, \mathbf{z}\) a fixed vector in \(\mathbb{R}^{n}\) i. \(T: \mathbf{P}_{n} \rightarrow \mathbf{P}_{n} ; T[p(x)]=p(x+1)\) j. \(T: \mathbb{R}^{n} \rightarrow V ; T\left(r_{1}, \cdots, r_{n}\right)=r_{1} \mathbf{e}_{1}+\cdots+r_{n} \mathbf{e}_{n}\) where \(\left\\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}\right\\}\) is a fixed basis of \(V\) k. \(T: V \rightarrow \mathbb{R} ; T\left(r_{1} \mathbf{e}_{1}+\cdots+r_{n} \mathbf{e}_{n}\right)=r_{1},\) where \(\left\\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}\right\\}\) is a fixed basis of \(V\)

Short answer

All functions T are linear transformations.

Step-by-step solution

Understanding Linear Transformations

To show that a function is a linear transformation, we need to verify two properties: 1) Additivity: \( T(u + v) = T(u) + T(v) \) for all vectors \(u, v\), and 2) Homogeneity: \( T(cu) = c T(u) \) for all scalars \(c\) and vectors \(u\).

Function a: Reflection in the x-axis

Consider \( T(x, y) = (x, -y) \). To check additivity, take \( u = (x_1, y_1) \) and \( v = (x_2, y_2) \). Then, \( T(u+v) = T(x_1+x_2, y_1+y_2) = (x_1+x_2, -(y_1+y_2)) = (x_1, -y_1) + (x_2, -y_2) = T(u) + T(v) \). Similarly, for homogeneity, \( T(cu) = T(cx_1, cy_1) = (cx_1, -cy_1) = c(x_1, -y_1) = cT(u) \). Thus, the function is a linear transformation.

Function b: Reflection in the x-y plane

Consider \( T(x, y, z) = (x, y, -z) \). For additivity, take \( u = (x_1, y_1, z_1) \) and \( v = (x_2, y_2, z_2) \). Then, \( T(u+v) = T(x_1+x_2, y_1+y_2, z_1+z_2) = (x_1+x_2, y_1+y_2, -(z_1+z_2)) = T(u) + T(v) \). For homogeneity, \( T(cu) = T(cx_1, cy_1, cz_1) = (cx_1, cy_1, -cz_1) = cT(u) \). Thus, \( T \) is a linear transformation.

Function c: Conjugation in Complex Numbers

For \( T(z) = \bar{z} \), choose any two complex numbers \( z_1 \) and \( z_2 \). Then additivity gives \( T(z_1 + z_2) = \overline{z_1 + z_2} = \bar{z_1} + \bar{z_2} = T(z_1) + T(z_2) \). For homogeneity, \( T(cz) = \overline{cz} = c\bar{z} = cT(z) \). This shows \( T \) is a linear transformation.

Function d: Matrix Multiplication by Fixed Matrices

Consider \( T(A) = PAQ \), where \( A \) is a matrix and \( P \), \( Q \) are fixed matrices. For additivity, \( T(A + B) = P(A + B)Q = (PAQ) + (PBQ) = T(A) + T(B) \). For homogeneity, \( T(cA) = P(cA)Q = c(PAQ) = cT(A) \). Thus, \( T \) is a linear transformation.

Function e: Symmetrical Sum of a Matrix and its Transpose

Given \( T(A) = A^T + A \), for matrices \( A \) and \( B \), additivity yields \( T(A+B) = (A + B)^T + (A + B) = A^T + B^T + A + B = T(A) + T(B) \). For homogeneity, \( T(cA) = (cA)^T + cA = cA^T + cA = c(A^T + A) = cT(A) \). Thus, \( T \) is a linear transformation.

Function f: Polynomial Evaluation at Zero

For \( T[p(x)] = p(0) \), take two polynomials \( p(x) \) and \( q(x) \). Then, \( T[p(x) + q(x)] = (p(0) + q(0)) = T[p(x)] + T[q(x)] \). For homogeneity, \( T[cp(x)] = cp(0) = cT[p(x)] \). Hence, this is a linear transformation.

Function g: Extracting the Leading Coefficient

With \( T(r_0 + r_1 x + \, ... \, + r_n x^n) = r_n \), take two polynomials. Additivity checks as \( T[p(x) + q(x)] = r_{n,p} + r_{n,q} = T[p(x)] + T[q(x)] \). Homogeneity is \( T[cp(x)] = cr_n = cT[p(x)] \). Thus, this is a linear transformation.

Function h: Dot Product with a Fixed Vector

Consider \( T(\mathbf{x}) = \mathbf{x} \cdot \mathbf{z} \) for a fixed \(\mathbf{z}\). Additivity gives \( T(\mathbf{u} + \mathbf{v}) = (\mathbf{u} + \mathbf{v}) \cdot \mathbf{z} = \mathbf{u} \cdot \mathbf{z} + \mathbf{v} \cdot \mathbf{z} = T(\mathbf{u}) + T(\mathbf{v}) \). Homogeneity: \( T(c\mathbf{x}) = (c\mathbf{x}) \cdot \mathbf{z} = c(\mathbf{x} \cdot \mathbf{z}) = cT(\mathbf{x}) \). Hence, \( T \) is a linear transformation.

Function i: Polynomial Shift by 1

Take \( T[p(x)] = p(x+1) \). For polynomials \( p(x) \) and \( q(x) \), \( T[p(x) + q(x)] = (p+q)(x+1) = p(x+1) + q(x+1) = T[p(x)] + T[q(x)] \). For homogeneity, \( T[cp(x)] = (cp)(x+1) = cT[p(x)] \). Thus, this is a linear transformation.

Function j: Linear Combination with Basis Vectors

For \( T(r_1, \, ... \, , r_n) = r_1 \mathbf{e}_1 + \, ... \, + r_n \mathbf{e}_n \), Given \( \mathbf{u} = (u_1, \, ..., \, u_n) \) and \( \mathbf{v} = (v_1, \, ..., \, v_n) \), additivity: \( T(\mathbf{u} + \mathbf{v}) = T((u_1+v_1), \, ..., \, (u_n+v_n)) = ((u_1+v_1)\mathbf{e}_1 + \, ... \, + (u_n+v_n)\mathbf{e}_n) = T(\mathbf{u}) + T(\mathbf{v}) \). Homogeneity: \( T(c\mathbf{u}) = T((cu_1), \, ..., \, (cu_n)) = ((cu_1)\mathbf{e}_1 + \, ... \, + (cu_n)\mathbf{e}_n) = cT(\mathbf{u}) \). Hence, \( T \) is a linear transformation.

Function k: Projection onto a Fixed Basis Component

For \( T(r_1 \mathbf{e}_1 + \, ... \, + r_n \mathbf{e}_n) = r_1 \), consider \( u = (r_1 \mathbf{e}_1 + \, ... \, + r_n \mathbf{e}_n) \) and \( v = (s_1 \mathbf{e}_1 + \, ... \, + s_n \mathbf{e}_n) \). Additivity yields \( T(u + v) = (r_1+s_1) = T(u) + T(v) \). Homogeneity holds as \( T(cu) = c(r_1) = cT(u) \). Thus, \( T \) is a linear transformation.

Key concepts

Additivity

The concept of additivity is one of the fundamental properties that a function must satisfy to be considered a linear transformation. This property requires that the function behaves predictably when applied to the sum of two vectors.
In mathematical terms, if we have a transformation function \( T: V \to W \), where \( V \) and \( W \) are vector spaces, then \( T \) is additive if it satisfies:\[ T(u + v) = T(u) + T(v) \] for every pair of vectors \( u, v \) in \( V \).
Think of this property as the ability of a function to "distribute" over vector addition.

• For example, consider the reflection function \( T(x, y) = (x, -y) \). If you take any two vectors \( u = (x_1, y_1) \) and \( v = (x_2, y_2) \), their sum \( u + v = (x_1 + x_2, y_1 + y_2) \) should satisfy \( T(u+v) = T(u) + T(v) \).

This formula means the resulting reflection of a vector sum is the same as the sum of the reflections of the individual vectors. This is crucial for many mathematical applications, ensuring consistency and predictability in how transformations apply to vector sums.

Homogeneity

Homogeneity, also known as the scalar multiplication property, ensures that linear transformations work in a proportionate way when the input vectors are scaled. In essence, this property ensures that if you scale a vector by a scalar, the result of the transformation scales in the same way.
Formally, for a transformation \( T: V \to W \), the transformation is homogeneous if it fulfills the condition: \( T(cu) = cT(u) \) for all vectors \( u \) in \( V \) and all scalars \( c \).

• Take, for example, the transformation \( T(x,y) = (x, -y) \) again. If you multiply \( u = (x_1, y_1) \) by some scalar \( c \), then \( cu = (cx_1, cy_1) \) should transform as \( T(cu) = cT(u) \), resulting in \( (cx_1, -cy_1) = c(x_1, -y_1) \).

Homogeneity conveys that the transformation respects the scaling factor, which is pivotal in maintaining the proportionality of relationships among vectors post-transformation. This is especially important in ensuring that transformations behave linearly, which is a core requirement in linear algebra.

Matrix Representation

Matrix representation is an essential concept in understanding linear transformations because matrices provide a compact and efficient way to represent these transformations.
Mathematically, if \( T: \, \mathbb{R}^n \to \mathbb{R}^m \) is a linear transformation, then there exists an \( m \times n \) matrix \( A \) such that for every vector \( x \) in \( \mathbb{R}^n \), the transformation is represented as: \( T(x) = Ax \).

• Consider a case where the transformation \( T \) involves the reflection of vectors across an axis, such as \( T(x, y) = (x, -y) \). This can be represented using a matrix \( A = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} \), so that \( T\left( \begin{bmatrix} x \ y \end{bmatrix} \right) = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} x \ -y \end{bmatrix} \).
• Matrix representation simplifies complex operations and assists in computations, such as finding the image of a vector under transformation or determining its inverse.

By using matrices, linear transformations can be easily manipulated and examined, allowing for deeper insights and simpler computations.