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How did Giuseppe Peano help establish the modern idea of vector spaces?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In 1888, the Italian mathematician Giuseppe Peano provided the first formal definition of vector spaces as abstract algebraic entities.1

Peano referred to these structures as "linear systems" because he recognized that any vector within such a space could be expressed as a linear combination of a finite set of vectors and scalars—represented by the form $a v + b w + \dots + c z$ . By establishing this framework, he formalized the concept that a set of vectors could generate the entire space, a collection now known as a spanning set. This work was foundational to linear algebra, as it allowed for the study of multidimensional quantities and their properties—such as addition and scalar multiplication—in an abstract, generalized way that preserved standard arithmetic properties.1