Final answer:
To find a basis for the vector space of all 3 x 3 symmetric matrices, you can consider the diagonal elements and use the matrices [1 0 0], [0 1 0], and [0 0 1]. For the vector space of all 3 x 3 skew-symmetric matrices, you can use the matrices [0 -1 0], [1 0 0], and [0 0 0].
Step-by-step explanation:
To find a basis for the vector space of all 3 x 3 symmetric matrices, we need to determine the linearly independent matrices that span this space.
One way to approach this is to consider the diagonal elements of the symmetric matrices. For example, a symmetric matrix can have the form:
[a b c]
[b d e]
[c e f]
Since the matrix is symmetric, the elements above and below the main diagonal are equal. Therefore, the basis for the vector space of all 3 x 3 symmetric matrices can be:
[1 0 0]
[0 1 0]
[0 0 1]
These three matrices are linearly independent and span the vector space.
To find a basis for the vector space of all 3 x 3 skew-symmetric matrices, we need to determine the linearly independent matrices that span this space.
A skew-symmetric matrix has the property that its transpose is equal to the negative of itself. For example, a skew-symmetric matrix can have the form:
[0 -a -b]
[a 0 -c]
[b c 0]
Since the matrix is skew-symmetric, the diagonal elements are all zero. Therefore, the basis for the vector space of all 3 x 3 skew-symmetric matrices can be:
[0 -1 0]
[1 0 0]
[0 0 0]
These three matrices are linearly independent and span the vector space.