Question

If \(z\) is a complex number, show that \(\left\\{z, z^{2}\right\\}\) is independent if and only if \(z\) is not real.

Short answer

\(z\) is not real, hence \(\{z, z^2\}\) is independent.

Step-by-step solution

Understand Complex Numbers

A complex number \(z\) can be expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. For \(z\) to be real, \(b = 0\), meaning \(z = a + 0i = a\).

Consider Linear Independence

The set \(\{z, z^2\}\) is said to be independent if there are no scalars \(c_1\) and \(c_2\), not both zero, such that \(c_1z + c_2z^2 = 0\). This implies \(z\) and \(z^2\) are linearly independent unless \(z^2\) is a scalar multiple of \(z\).

Analyze When \(z^2\) Is a Multiple of \(z\)

For \(z^2\) to be a multiple of \(z\), we must have \(z^2 = cz\) where \(c\) is a scalar. Simplifying gives \(z(z - c) = 0\). This equation implies that either \(z = 0\) or \(z - c\) is a scalar multiple of itself only when \(z = c\).

Examine Conditions for Real \(z\)

If \(z\) is real (\(z = a\)), then \(z^2 = a^2\) is clearly a multiple of \(z = a\), showing dependence. Thus, the set becomes dependent when \(z\) is real.

Examine Conditions for Non-Real \(z\)

If \(z\) is not real, \(z = a + bi\) with \(b eq 0\), then \(z^2 = (a^2 - b^2) + 2abi\) is not a scalar multiple of \(z = a + bi\), ensuring independence.

Key concepts

Linear Independence

When dealing with sets in mathematics, such as \( \{z, z^2\} \) where \( z \) is a complex number, it's crucial to determine whether these elements are linearly independent. What does this mean in simple terms? Linear independence implies that no element in the set can be written as a linear combination of the others. In this context, we need to make sure there are no scalars \( c_1 \) and \( c_2 \), not both zero, for which \( c_1z + c_2z^2 = 0 \).

This condition can be checked by testing if \( z^2 \) can be expressed as a scalar multiple of \( z \). If \( z^2 = cz \) for some scalar \( c \), then \( \{z, z^2\} \) is dependent. This exercise shows us that dependency happens when a complex number is real, which turns out to be a special case.

Real and Imaginary Parts

To understand complex numbers, we break them down into their real and imaginary components. A complex number is generally expressed as \( z = a + bi \), where \( a \) and \( b \) are real numbers. Here, \( a \) is called the real part, while \( b \) is the imaginary part.

For a number to be strictly real, we set \( b = 0 \), thus making \( z = a \). Real numbers and complex numbers have distinct characteristics, especially when considering linear independence. If \( z \) is real, then \( z^2 = a^2 \) can be rewritten as a multiple of \( z \), showing dependency. But if \( b eq 0 \), then the complex number has a nature that usually doesn't satisfy the scalar multiplication condition, ensuring independence.

Scalar Multiplication

Scalar multiplication involves multiplying a complex number by a real number, or scalar. When we multiply \( z = a + bi \) by a scalar \( c \), the result is \( cz = ca + cbi \).

Understanding scalar multiplication is vital in assessing linear independence. If a complex number, when squared, remains a scalar multiple of the original number, this suggests dependence, because both numbers are bound by a simple scalar relation. However, when \( z \) is a non-real complex number, the squaring alters its form significantly, meaning it generally does not reduce simply to \( cz \), thus confirming independence.

To sum it up, scalar multiplication helps us understand dependencies and independencies in the set \( \{z, z^2\} \) by clarifying when one part can or cannot be expressed as a multiple of the other.