Question

Which of the following is/are linear transformations ?

• A. T : \(\R → \R\) given by T(x) = sin(x)
• B. T : \(M_2(\R) → \R\) given by T(A) = trace(A)
• C. T : \(\R^2 → \R\) given by T (x, y) = x + y + 1
• D. T : \(P_2(\R) → \R\) given by T(p(x)) = p(1)

Answer 1

The Correct Option is B, D

To determine which of the given transformations are linear, we must consider the definition of a linear transformation. A transformation \( T: V \to W \) is linear if for all vectors \( \mathbf{u}, \mathbf{v} \) in \( V \) and all scalars \( c \), the following properties hold:

1. Additivity: \( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \)
2. Homogeneity: \( T(c\mathbf{u}) = cT(\mathbf{u}) \) 

Let's analyze each transformation:

1. Transformation \( T: \mathbb{R} \to \mathbb{R} \) given by \( T(x) = \sin(x) \):

This transformation is not linear because it does not satisfy the homogeneity property. For instance, \( T(cx) = \sin(cx) \neq c\sin(x) \), hence, it doesn't satisfy multiplicative scaling.

1. Transformation \( T: M_2(\mathbb{R}) \to \mathbb{R} \) given by \( T(A) = \text{trace}(A) \):

The trace of a matrix \( A \) is defined as the sum of its diagonal entries. For matrices \( A \) and \( B \), and scalar \( c \), the properties hold:

• \( T(A + B) = \text{trace}(A + B) = \text{trace}(A) + \text{trace}(B) = T(A) + T(B) \)
• \( T(cA) = \text{trace}(cA) = c\text{trace}(A) = cT(A) \)

1. Transformation \( T: \mathbb{R}^2 \to \mathbb{R} \) given by \( T(x, y) = x + y + 1 \):

This transformation is not linear because of the constant term \( +1 \). For linearity, the transformation must map the zero vector to zero, but here \( T(0, 0) = 1 \neq 0 \), indicating it is not a linear transformation.

1. Transformation \( T: P_2(\mathbb{R}) \to \mathbb{R} \) given by \( T(p(x)) = p(1) \):

This transformation takes a polynomial and evaluates it at \( x = 1 \). For polynomials \( p(x) \) and \( q(x) \), and scalar \( c \), the linearity properties can be checked as follows:

• \( T(p(x) + q(x)) = (p + q)(1) = p(1) + q(1) = T(p(x)) + T(q(x)) \)
• \( T(cp(x)) = cp(1) = cT(p(x)) \)

Conclusion: The linear transformations among the given options are:

• \( T: M_2(\mathbb{R}) \to \mathbb{R} \) given by \( T(A) = \text{trace}(A) \)
• \( T: P_2(\mathbb{R}) \to \mathbb{R} \) given by \( T(p(x)) = p(1) \)