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 $2 \times 3$ matrices with all entries nonegative.

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Key Concepts

Vector Space

A vector space is a set equipped with two operations, vector addition and scalar multiplication, that satisfy certain axioms such as associativity, commutativity, the existence of an additive identity and inverses, and distributivity. A critical requirement is that the set must be closed under these operations, meaning that performing these operations on elements of the set should always result in an element that is also in the set.

Closure Under Scalar Multiplication

This concept is essential for determining if a set qualifies as a vector space. It requires that, for any vector in the set and any scalar in the field, the product of the scalar and the vector must remain within the set. If a constraint on the set (such as nonnegativity of entries) is not preserved under multiplication by arbitrary scalars (especially negative ones), the set will not be closed and therefore cannot be a vector space.

Basis and Dimension

A basis of a vector space is a set of linearly independent vectors that spans the whole space, meaning every vector in the space can be expressed as a linear combination of the basis elements. The dimension of the vector space is defined as the number of vectors in any basis, providing a measure of the 'size' or 'complexity' of the space in terms of independent directions.

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