AcademySubspaces And Dimension
Academy
Dimension In Rn
Level 1 - Linear Algebra topic page in Subspaces And Dimension.
Principle
The dimension of a subspace is the number of vectors in any basis for that subspace. Dimension counts the number of independent directions needed to describe every vector in the subspace.
All bases of the same subspace have the same number of vectors.
Notation
\(\dim W\)
the dimension of subspace W
\(B\)
a basis
\(\mathbb R^n\)
n-dimensional real coordinate space
\(\operatorname{span}(B)\)
the subspace spanned by basis B
\(\{\mathbf0\}\)
the zero subspace containing only the zero vector
The Core Method
To find the dimension of a subspace, find a basis and count its vectors:
Dimension from a basis
\[B\text{ is a basis for }W\Longrightarrow \dim W=|B|\]
Important standard dimensions are
Coordinate-space dimension
\[\dim\mathbb R^n=n\]
and
Zero-space dimension
\[\dim\{\mathbf0\}=0\]
Geometrically in \(\mathbb R^3\): a line through the origin has dimension \(1\), a plane through the origin has dimension \(2\), and all of \(\mathbb R^3\) has dimension \(3\).
Worked Cases
Question
Find the dimension of \(W=\{(x,y,0):x,y\in\mathbb R\}\).
Answer
A basis for \(W\) is \(((1,0,0),(0,1,0))\). These two vectors span every vector \((x,y,0)\) by \(x(1,0,0)+y(0,1,0)\), and they are independent. Since the basis has \(2\) vectors, \(\dim W=2\).
Examples
Question
What is \(\dim\mathbb R^5\)?
Answer
The standard basis of \(\mathbb R^5\) has five vectors: \(\mathbf e_1,\mathbf e_2,\mathbf e_3,\mathbf e_4,\mathbf e_5\). Therefore \(\dim\mathbb R^5=5\).