Final answer:
To find a spanning set for the symmetric 3x3 space, start with the standard basis vectors and perform operations such as addition and subtraction to generate more vectors.
Step-by-step explanation:
A spanning set is a set of vectors that, when combined, can form any vector in a given vector space. In the case of a symmetric 3x3 space, we can represent it as a matrix. To find a spanning set, we need to find vectors that can form any 3x3 symmetric matrix.
One way to do this is to start with the standard basis vectors {(1, 0, 0), (0, 1, 0), (0, 0, 1)}. These vectors are linearly independent and can form the standard basis for a 3x3 matrix.
To generate more vectors, we can perform operations on the standard basis vectors such as addition, subtraction, and scalar multiplication. For example, we can add the first and second basis vectors to get (1, 1, 0). Similarly, we can subtract the first and third basis vectors to get (1, 0, -1). These vectors along with the standard basis vectors form a spanning set for the symmetric 3x3 space.