These notes give the definition of a vector space and several of the concepts related to these spaces. Examples are drawn from the vector space of vectors in .
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A nonempty set of elements is called a vector space if in there are two algebraic operations (called addition and scalar multiplication), so that the following properties hold.
Addition associates with every pair of vectors and a unique vector which is called the sum of and and is written . In the case of the space of 2-dimensional vectors, the summation is componentwise (i.e. if and , then ), which can be best illustrated by the ``parallelogram illustration'' below:
Addition satisfies the following :
The use of an additive inverse allows us to define a subtraction operation on vectors. Simply The result of vector subtraction in the space of 2-dimensional vectors is shown below.
Frequently this 2-d vectors is protrayed as joining the ends of the two original vectors. As we can see, since the vectors are determined by direction and length, and not position, the two vectors are equivalent.
Scalar Multiplication associates with every vector and every scalar , another unique vector (usually written ),
For scalar multiplication the following properties hold:
Examples of vector space abound in mathematics. The most obvious examples are the usual vectors in , from which we have drawn our illustrations in the sections above. But we frequently utilize several other vectors spaces: The 3-d space of vectors, the vector space of all polynomials of a fixed degree, and vector spaces of matrices. We briefly discuss these below.
The Vector Space of 3-Dimensional Vectors
The vectors in also form a vector space, where in this case the vector operations of addition and scalar multiplication are done componentwise. That is and are vectors, then addition is
The axioms are easily verified (for example the additive identity of is just , and the zero vector is just . Here the axioms just state what we always have been taught about these sets of vectors.
Vector Spaces of Polynomials
The set of quadratic polynomials of the form
The axioms are again easily verified by performing the operations individually on like terms.
A simple extension of the above is to consider the set of polynomials of degree less than or equal to . It is easily seen that these also form a vector space.
Vector Spaces of Matrices
The set of Matrices form a vector space. Two matrices can be added componentwise, and a matrix can be multiplied by a scalar. All axioms are easily verified.
Given a vector space , the concept of a basis for the vector space is fundamental for much of the work that we will do in computer graphics. This section discusses several topics relating to linear combinations of vectors, linear independence and bases.
This element is clearly a member of the vector space (just repeatedly apply the summation and scalar multiplication axioms).
The set that contains all possible linear combinations of is called the span of . We frequently say that is spanned (or generated) by those vectors.
It is straightforward to show that the span of any set of vectors is again a vector space.
Given a set of vectors from a vector space . This set is called linearly independent in if the equation
If a set of vectors is not linearly independent, then it is called linearly dependent. This implies that the equation above has a nonzero solution, that is there exist which are not all zero, such that
Any set of vectors containing the zero vector ( ) is linearly dependent.
To give an example of a linear independent set that everyone has seen, consider the three vectors
Consider the equation
Let be a set of vectors in a vector space and let be the span of . If is linearly independent, then we say that these vectors form a basis for and has dimension . Since these vectors span , any vector can be written uniquely as
If is the entire vector space , we say that forms a basis for , and has dimension .