### Basis and dimension

**Definition 1**. We say that vectors form a **basis** in a subspace if 1) it is spanned by and 2) these vectors are linearly independent. The number of vectors in the basis is called a **dimension** of and the notation is

An orthogonal basis is a special type of a basis, when in addition to the above conditions 1)-2) the basis vectors are orthonormal. For the dimension definition to be correct, the number of vectors in any basis should be the same. We prove correctness in a separate post.

**Exercise 1**. In the unit vectors are linearly independent. Prove this fact 1) directly and 2) using the properties of an orthonormal system.

**Direct proof**. Any can be represented as

(1)

If the right side is zero, then and all are zero.

**Proof using orthonormality**. If the right side in (1) is zero, then for all

**Exercise 2**.

**Proof**. (1) shows that is spanned by Besides, they are linearly independent by Exercise 1.

**Definition 2**. Let be two subspaces such that any element of one is orthogonal to any element of the other. Then the set is called an **orthogonal sum** of and denoted

**Exercise 3**. If a vector belongs to both terms in the orthogonal sum of two subspaces , then it is zero. This means that

**Proof**. This is because any element of is orthogonal to any element of so is orthogonal to itself, and

**Exercise 4 **(*dimension additivity*) Let be an orthogonal sum of two subspaces. Then

**Proof**. Let By definition, is spanned by some linearly independent vectors and is spanned by some linearly independent vectors Any can be decomposed as Since can be further decomposed as the system spans

Moreover, this system is linearly independent. If

then

By Exercise 3 then By linear independence of the vectors in the two systems all coefficients must be zero.

The conclusion is that