This is a brief introduction to concepts in linear algebra to support study of subjects such as spectral clustering in bioinformatics. For more detail you will be directed to web pages in the Wikipedia and in the Khan Academy.
The major topics are:
A vector is an ordered list of numbers, e.g. [1, 5, -7]. The numbers may be real numbers, complex numbers, quaternions, or other type of numbers that we can add, subtract or multiply using the usual associative and distributive rules of arithmetic. Multiplication need not be commutative, e.g. when working with quaternions.
If the number of elements in the list is called the dimension of the vector, and each element is called a component of the vector. We add or subtract vectors of the same dimension component by component, so that [1, 3, 5] + [2, 4, 6] adds up to [3, 7, 11]. We call a single number by itself a scalar. We multiply a scalar by a vector by pre-multiplying each vector component by the scalar. We can also form the dot product of two vectors of the same dimension by multiplying component by component and adding the results to form a scalar, so that [1, 3, 5].[2, 4, 6] = 2 + 12 + 30 = 44.
All the vectors of a given dimension using a particular set of numbers is call a vector space, written as the symbol for that set of numbers with a superscript giving the dimension. For example, the space of 3-dimensional vectors over the real numbers is written as R3 and the space of 6-dimensional vectors over the complex numbers is written as C3.
For more in vector spaces, see the wikipedia article https://en.wikipedia.org/wiki/Vector_space and the Khan Academy module https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces