A vector space is when you can:

• add two Things
• multiply a Thing by any real number

And get another Thing that’s the same Kind of Thing.

By Thing I mean Vector and by Kind of Thing I mean element of the same Vector Space.

Examples of vector spaces:

• real numbers
• complex numbers
• sets of N numbers (what most people think of when they hear “vector”)
• matrices
• polynomials
• functions
• quantum states of a given system
• quantities of apples sold, classified by type of apple

Examples of Not Vector Spaces:

• integers
• negative numbers
• nonzero numbers
• unitary matrices
• apples

Yeah a few of these come with asterisks I’m happy to answer questions but don’t want to argue with pedants.