Linear algebra is essential the study of vector spaces and their transformations. A vector space is a mixed object comprising two constituents: vectors and scalars.
Scalars behave like rational numbers. They can be added, subtracted and multiplied. Division by any non-zero scalar is also possible. In short, they form a field
Vectors can be added and subtracted but not, in general, multiplied by each other. They form an abelian group.
The interaction between vectors and scalars consists of ``multiplying'' a vector by a scalar.
To compare vector spaces we have the notion of a linear transformation, which is just a function between vector spaces with common scalars such that the function respects the vector space structure. We consider two vector spaces with common scalars to be essentially the same, or isomorphic, if there is a linear transformation between them which has an inverse. (We shall see that any inverse function is automatically a linear function.) Intuitively, two vector spaces with common scalars are isomorphic if the only difference between them is what the elements are called, or how they are designated. It is in this sense that they are essentially the same.
One of the tasks of linear algebra is to classify vector spaces with common scalars. It is an amazing fact that the question as to whether two vector spaces with common scalars are isomorphic is decidablee by means of a single numerical invariant, the dimension: Two vector spaces with common scalars are isomorphic if and only if they have the same dimension. This is proved by showing that each vector space has a basis and two vector spaces with common scalars are isomorphic if and only if any basis for one has the same number of elements as any basis for the other. This number is the dimension of the vector space.
All of this holds for all vector spaces.
If we choose a basis for each of our vector spaces, and if each has finite dimension, then we can represent each vector by a coordinate vector and each linear transformation by a matrix whose coefficients are scalars. Composition of linear transformations is then represented by multiplication of matrices. It is this use of matrices which makes many things computable. For example, if the matrix has non-zero determinant, then it represents an isomorphism, and vice-versa.
We shall also see how to construct new vector spaces from given ones. In particular, we shall meet the direct sum of vector spaces with common scalars and we shall see when a subset of a vector space forms a vector subspace.
In fact, every vector space can be written as the direct sum of non-trivial subspaces, none of which can be further decomposed in this manner, their number being precisely the dimension of the vector space: Each indecomposable summand has dimension one.
We can similarly form direct sums of linear transformations. But while every vector space is a direct sum of as many one-dimensional subspaces as its dimension, it is not true in general that every linear transformation of a vector space into itself can be expressed as the direct sum of that many linear transformations between such subspaces! Indeed, this is possible if and only if the vector space has a basis consisting of eigenvectors. Then (in the finite-dimensional case) the only possible non-zero entries in the matrix of the linear transformation with respect to such a basis, are the diagonal ones, which are precisely the eigenvalues. Eigenvalues and eigenvectors are central to applications of linear algebra.
All this is true of any vector space with, at most, the
restriction to finite dimensionality.
Some vector spaces admit additional structure. One such structure is an inner product. This allows us to study the vector space in question geometrically. The inner product allows us to measure angles and speak of distances in the vector space concerned. This richer structure allows for a deeper study and finds wide application both within mathematics and in other sciences -- both natural and social -- and technology. One application which should be familiar is digital recording of sound.