Question

Find the dimension of the subspace \(\operatorname{span}\left\\{1, \sin ^{2} \theta, \cos 2 \theta\right\\}\) of \(\mathbf{F}[0,2 \pi]\).

Short answer

The dimension of the subspace is 2.

Step-by-step solution

Identify the Elements of the Set

We are given the set consisting of functions: \(\{1, \sin^2 \theta, \cos 2\theta\}\). The goal is to discover a minimal linearly independent subset that spans the same subspace in \(\mathbf{F}[0, 2\pi]\), the space of functions defined on the interval \([0, 2\pi]\).

Express Functions in Terms of Trigonometric Identities

Use trigonometric identities to express the elements in terms of more common functions. We know \(\sin^2 \theta = \frac{1 - \cos 2\theta}{2}\). Substitute this into the set to get substituting expressions \(\{1, \frac{1 - \cos 2\theta}{2}, \cos 2\theta\}\).

Determine Potential Linear Combinations

Use linear combinations to examine if any function can be expressed as a combination of the others. Notice that \(\cos 2\theta\) and its identified equivalent \(\frac{1 - \cos 2\theta}{2}\) already suggest a dependency.

Solve Linear Dependencies

Consider the functions \(1\) and \(\cos 2\theta\) are independent, and check if \(\sin^2 \theta\) can be written as a linear combination of these two. Using the identity, write \(\sin^2 \theta = \frac{1}{2} - \frac{1}{2} \cos 2\theta\), which shows dependency.

Determine Linear Independence of Reduced Set

The potential set of functions \(\{1, \cos 2\theta\}\) remains. Test for independence by assuming a linear relationship \(a \cdot 1 + b \cdot \cos 2\theta = 0\) must hold for all \(\theta\). This implies \(a = 0\) and \(b = 0\), confirming independence.

Conclude the Dimension

Since the reduced set \(\{1, \cos 2\theta\}\) is linearly independent and spans the equivalent subspace, and has 2 elements, the dimension of the subspace is 2.

Key concepts

Subspaces

In linear algebra, a subspace is a subset of a vector space that is closed under addition and scalar multiplication. This means if you have two elements in a subspace and you add them, or multiply them by a scalar, the result will still be in the subspace. In our exercise, the subspace is the span of the set \( \{1, \sin^2 \theta, \cos 2\theta\} \) within the space of functions defined over \([0, 2\pi]\). This subset forms a new "space" of functions that include combinations of these basic functions.

Linear Independence

Linear independence is a way of determining whether a set of vectors (or functions) in a vector space are independent of each other. A set is linearly independent if no vector in the set can be written as a combination of the others. To test for this, we set up equations to see if coefficients can only have a trivial solution (where all coefficients are zero). In the given exercise, the goal was to find such an independent set by expressing \( \sin^2 \theta \) in terms of \( 1 \) and \( \cos 2\theta \). Successfully doing this with all zero coefficients confirms the set's independence.

Trigonometric Identities

Trigonometric identities help simplify complex trigonometric expressions into simpler ones. They are crucial in our exercise by transforming \( \sin^2 \theta \) into another form using the identity \( \sin^2 \theta = \frac{1 - \cos 2\theta}{2} \). This enables us to see how these functions relate to each other. Using identities in different contexts can simplify problems in linear algebra, making it easier to find dependencies and simplify expressions.

Spanning Sets

A spanning set for a space means that any element of that space can be derived as a combination of elements of this set. In simpler terms, the set gives you all the tools you need to "build" any vector in the space. In the exercise, the original set \( \{1, \sin^2 \theta, \cos 2\theta\} \) spans a subspace of \( \mathbf{F}[0,2\pi] \). By reducing it to the independent spanning set \( \{1, \cos 2\theta\} \), we maintain the span while ensuring linear independence. This helps determine the dimension, which in this case is 2.