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I am wondering why is it that matrices and infinite sequences may be considered part of a vector space. I have read 3 different sources, and my interpretation of a vector space is something that belongs in a field and follows a list of properties that are standard to real numbers, i.e association, commutativity, zero property etc. It must have closure by addition and scalar multiplication, as well as being a nonempty set.

Is the reason that a matrix can be included in a vector space is that it can be multiplied to a vector to give a constant. I think this would make sense since matrices follow the properties listed above, but how linear equations exist in a real number space pervades me, perhaps it is similar to a straight line existing in an xyz-coordinate system.

Ax = b where x is a vector

How is a vector space different from a typical coordinate system, other than it can go into higher dimensions?

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# Vector spaces

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