Find a basis for each of these subspaces of 3 by 3 matrices:...

Find a basis for each of these subspaces of 3 by 3 matrices:
(a) All diagonal matrices.
(b) All symmetric matrices $\left(A^{\mathrm{T}}=A\right)$.
(c) All skew-symmetric matrices $\left(A^{T}=-A\right)$.

Key Concepts

Symmetric Matrices

A symmetric matrix is one that is equal to its transpose, meaning that the entry in the i-th row and j-th column is equal to the entry in the j-th row and i-th column. The space of symmetric matrices is characterized by independent entries along the diagonal and in either the upper or lower triangular part, with the other half determined by symmetry.

Vector Spaces and Subspaces

A vector space is a collection of objects called vectors which can be added together and multiplied by scalars, and a subspace is a subset of a vector space that is itself a vector space under the same operations. In the context of matrices, various subsets defined by specific properties (like being diagonal, symmetric, or skew-symmetric) are subspaces.

Basis

A basis of a vector space or subspace is a set of linearly independent elements that span the entire space, meaning that every element of the space can be written uniquely as a linear combination of the basis elements. Choosing an appropriate basis simplifies the study and representation of the subspace.

Diagonal Matrices

A diagonal matrix is a square matrix that has nonzero entries only along the main diagonal, with all off-diagonal entries being zero. The subspace of diagonal matrices is spanned by matrices that each have a single 1 in one diagonal position and 0 in all other positions, allowing any diagonal matrix to be expressed as a linear combination of these.

Skew-Symmetric Matrices

A skew-symmetric matrix is defined by the property that its transpose is the negative of the original matrix. Consequently, all diagonal entries must be zero and the off-diagonal entries come in pairs as negatives of each other. This property restricts the number of independent parameters and thus defines the subspace of skew-symmetric matrices.

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