Question

Show that the following transformations are linear: (a) \(V\) is \(\mathbb{C}\) over the reals and \(\mathbf{C}|z\rangle=\left|z^{*}\right\rangle .\) Is \(\mathbf{C}\) linear if instead of real numbers, complex numbers are used as scalars? (b) \(\quad V\) is \(\mathfrak{P}^{c}[t]\) and \(\mathbf{T}|x(t)\rangle=|x(t+1)\rangle-|x(t)\rangle\).

Short answer

The transformation \(C\), operating on complex vectors with real scalars, is indeed linear because it satisfies the conditions of additivity and homogeneity. However, when \(\mathbb{C}\) is used as the scalar, \(C\) is not linear anymore. For the transformation \(T\), on the polynomial vector space, it is linear, satisfying both conditions of additivity and homogeneity.

Step-by-step solution

Testing the transformation \(C\) for additivity over the reals

For two complex numbers \(z_1\) and \(z_2\), we have to check if \(C(z_1 + z_2) = C(z_1) + C(z_2)\). If transformation \(C\) satisfies this equation, then \(C\) is additive.

Testing the transformation \(C\) for homogeneity over the reals

Next, for a real number \(a\) and a complex number \(z\), check if \(C(az) = aC(z)\). If transformation \(C\) satisfies this equation, then \(C\) is homogeneous.

Assessing the linearity of \(C\) with complex scalars

Now, let's look at what happens when a complex number is used as a scalar instead. Repeat the steps from above to check for additivity and homogeneity, but this time with \(a \in \mathbb{C}\).

Testing the transformation \(T\) for additivity

Now we need to prove if the transformation \(T\) is linear. Start by testing additivity. For two polynomials \(x_1(t)\) and \(x_2(t)\) we must verify whether \(T(x_1(t) + x_2(t)) = T(x_1(t)) + T(x_2(t))\). If \(T\) satisfies this equation, then it is additive.

Testing the transformation \(T\) for homogeneity

Check homogeneity by choosing any real number \(a\), and polynomial \(x(t)\). We must verify if \(T(ax(t)) = aT(x(t))\). If this equation holds, then \(T\) is homogeneous and therefore linear.

Key concepts

Additivity

Additivity is a key characteristic of linear transformations. It means that a transformation respects the addition of elements. To check if a transformation like \( C \) is additive, you must verify whether the transformed sum of two elements equals the sum of their separate transformations. For example, if you have two complex numbers \( z_1 \) and \( z_2 \), the transformation \( C \) should satisfy the condition:

• \( C(z_1 + z_2) = C(z_1) + C(z_2) \)

This ensures that \( C \) maintains the structure of addition within the set. Additivity is essential because it tells us that a linear transformation, when dealing with multiple inputs, will treat them consistently, maintaining the underlying algebraic structure.

Homogeneity

Homogeneity in linear transformations refers to the ability to consistently scale elements. If you multiply an element by a scalar, the transformation should also be scaled by that factor. For a transformation \( C \), homogeneity over the reals requires that for any real scalar \( a \) and for a complex number \( z \), the equation aligns as:

• \( C(az) = aC(z) \)

Similarly, for a polynomial transformation \( T \), homogeneity means applying this concept to polynomials, which can be verified with real numbers. Essentially, homogeneity ensures that scaling phenomena are reliably observed. This is crucial because it aligns with the intuitive understanding of proportionality and scalability in mathematics, which is foundational to linearity.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, typically expressed as \( a + bi \) where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit satisfying \( i^2 = -1 \). In the context of linear transformations, understanding how transformations behave under complex operations is essential for comprehending their full behavior. For instance:

• How do linear properties change if scalars (multiplying or adding) are complex instead of real?
• Does the transformation maintain its linearity when these complex components are manipulated?

For the transformation \( C \) mentioned earlier, it becomes important to test if replacing real with complex scalars maintains properties like additivity and homogeneity. This exploration helps clarify the true nature and scope of the transformation in the complex domain.

Polynomials

Polynomials are expressions consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Consider the transformation \( T \) acting on polynomials. Checking if it's linear involves validating additivity and homogeneity:

• For additivity, you check if applying \( T \) to a sum of polynomials adheres to the transformation's structure.
• For homogeneity, scaling a polynomial by a scalar and then transforming should be equivalent to transforming the polynomial first and then scaling the result.

This systematic testing ensures that the transformation respects the polynomial structure at each step. Polynomials are significant in linear algebra due to their wide range of applications, from simple algebraic expressions to complex systems analysis and design.

Real Numbers

Real numbers encompass all the numbers on the number line, including both rational and irrational numbers. They are commonly used to scale and add elements in transformations. Understanding transformations over real numbers gives insight into basic linear operators. For transformations like \( C \) and \( T \) to be linear, they must demonstrate additivity and homogeneity over the reals.

• Real scalars are often what we initially use to simplify testing of linear properties due to the unvarying nature of real numbers.
• This foundational understanding is crucial before delving into more complex scalar types, like complex numbers.

Recognizing how real numbers interact with transformations helps build the backbone of algebraic frameworks, essential for tackling advanced mathematical concepts.