One important property of orthogonal sets is that they are linearly independent. To see why, suppose that we have a linear combination of the vectors in the set that equals zero: c_1 v_1 + c_2 v_2 + ... + c_n v_n = 0. Taking the dot product of both sides with v_i, we get c_i ||v_i||^2 = 0, where ||v_i|| represents the length of the vector v_i. Since the vectors are not zero, ||v_i|| ≠ 0, so we must have c_i = 0. Thus, the only linear combination that equals zero is the trivial one, which means that the vectors in the set are linearly independent. Another important property of orthogonal sets is that they can be used to form an orthonormal set, which is a set of vectors that are orthogonal and have unit length. To obtain an orthonormal set from an orthogonal set, we simply divide each vector by its length: {v_1/||v_1||, v_2/||v_2||, ..., v_n/||v_n||}. This set is still orthogonal, since the dot product of any two distinct vectors is zero, and it is orthonormal, since the length of each vector is one. An orthogonal basis is a basis of a vector space that is also an orthogonal set. That is, the basis vectors are mutually perpendicular to each other. One important property of orthogonal bases is that they are automatically orthonormal, since we can simply divide each vector by its length to obtain a unit vector. Orthogonal bases have many applications in linear algebra. For example, they can be used to find the coordinates of a vector with respect to a given basis. Suppose we have an orthogonal basis {v_1, v_2, ..., v_n} for a vector space V. Then, any vector v ∈ V can be written as a linear combination of the basis vectors: v = c_1 v_1 + c_2 v_2 + ... + c_n v_n. To find the coefficients c_i, we can take the dot product of both sides with v_i: v · v_i = (c_1 v_1 + c_2 v_2 + ... + c_n v_n) · v_i = c_i ||v_i||^2, since the dot product of any two distinct basis vectors is zero. Therefore, c_i = (v · v_i)/||v_i||^2. This formula allows us to compute the coordinates of v with respect to the basis {v_1, v_2, ..., v_n}. Orthogonal bases also have applications in diagonalization. Suppose we have a linear transformation T : V → V that is represented by a matrix A with respect to an orthogonal basis {v_1, v_2, ..., v_n}. Then, the matrix A is also orthogonal, since the columns of A are the coordinates of the basis vectors with respect to the standard basis. Since A is orthogonal, its inverse is its transpose: A^T. Therefore, we can diagonalize A by finding an orthogonal basis of