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How to prove something is a basis

Let B = { 1, cos t, cos 2 t, cos 3 t, cos 4 t } and C = { 1, cos t, cos 2 t, cos 3 t, cos 4 t } and assume the following identities: Let H = S p a n { B }. Prove that C is another basis for H by

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I am still confused how to prove that a set is a basis other than

The standard way of proving that something is a basis is to prove that it is linear independent and that it spans the vector space. Of course, sometimes there are shortcuts.

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Clarify mathematic equation

Mathematics is the study of numbers, shapes, and patterns. It is used to describe and explain the physical world around us.

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How to check if a set of vectors is a basis

In mathematics, a set B of vectors in a vector space V is called a basis if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of
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