Is the given set (taken with the usual addition and scalar...

Is the given set (taken with the usual addition and scalar multiplication) a vector space? (Give a reason.) If your answer is yes, find the dimension and a basis.All symmetric $3 \times 3$ matrices.

Key Concepts

Vector Space

A vector space is a collection of objects called vectors that can be added together and multiplied by scalars while obeying certain axioms (such as associativity, commutativity, existence of an additive identity and inverses, and distributive properties). In the context of matrices, this means that the set must be closed under matrix addition and scalar multiplication.

Closure Properties

For any subset of a vector space to itself be a vector space (or subspace), it must be closed under the operations of addition and scalar multiplication. This ensures that when you add any two elements from the set or multiply any element by a scalar, the result remains within the set.

Symmetric Matrix

A symmetric matrix is one that is equal to its own transpose; that is, the entry in the ith row and jth column is equal to the entry in the jth row and ith column. This property imposes specific requirements on the entries of the matrix and is key to understanding the structure of the set being considered.

Basis and Dimension

A basis of a vector space is a set of linearly independent vectors that spans the entire space. The dimension is the number of vectors in any basis for the space. Identifying a basis and determining the dimension are fundamental steps in understanding the structure and properties of a vector space, such as the space of symmetric matrices.

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