Basis of Dimension
- (Harvard University - Harvard Taiwan Student Association)
- Overview
In linear algebra, a basis is a linearly independent set of vectors that spans a subspace. The number of vectors in a basis is called the dimension of the subspace.
Here are some examples of dimensions:
- Rn: The dimension is n.
- Vector space of polynomials in x: The dimension is 3 if the polynomials have real coefficients and a degree of at most two.
- Vector space that consists of only the zero vector: The dimension is zero.
A vector space can have multiple bases, but all bases have the same number of elements. If there is no finite basis, the vector space is called infinite dimensional. Otherwise, the vector space is called finite dimensional.
A basis can be found by rewriting a subspace as a column space or a null space. For example, the pivot columns of a matrix form a basis for the column space of the matrix.
- Linear Algebra in Three Dimensions
Linear algebra is the study of linear combinations, vector spaces, lines, and planes. It also includes linear functions and matrices.
In three dimensions, linear algebra is used in 3D graphics programming. It can represent points in 3D space, move points between coordinate frames, and remove the third dimension when projecting points onto the screen.
Linear algebra is also used in real life to determine unknown quantities. Some real-life applications of linear algebra include:
Calculating speed, distance, or time
Projecting a three-dimensional view into a two-dimensional plane
A three-dimensional linear transformation is a function of the form:
T(x,y,z)=(a11x+a12y+a13z,a21x+a22y+a23z,a31x+a32y+a33).
A linear transformation with three-dimensional vectors involves moving points around in three-dimensional space while keeping grid lines parallel and evenly spaced, and fixing the origin in place.
- Projecting a 3-Dimensional view into a 2-Demensional Plane
In linear algebra, a plane is two-dimensional because every point on the plane can be described by a linear combination of two independent vectors.
In three-dimensional space, a plane can be defined by three points that are not on the same line. A 3D linear subspace can be either a 3D space or a 2D plane within a 3D space. A 3D linear subspace is a 3D space if the three vectors are not coplanar, meaning they do not lie on the same plane. If the three vectors are all on the same plane, the 3D linear subspace is a 2D plane.
In three dimensions, coordinate planes are defined by the coordinate axes. There are three axes, so there are three intersecting pairs of axes. Each pair of axes forms a coordinate plane: the xy-plane, the xz-plane, and the yz-plane.
[More to come ...]
