Rank/Dimension of a Vector Space

<aside> 💡 The dimension(or rank) of a vector space is the size(or cardinality) of a basis of the vector space.

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For every vector space there exists a basis, and all bases of a vector space have the same number of elements(or cardinality).

<aside> 💡 The dimension(or rank) of a vector space (say $V)$ is uniquely defined and denoted by $dim(V)$ (or $rank(V)$) respectively.

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Dimension of $\R^n$

The $i^{th}$ standard basis vector in $\R^n$:

$e_i=(0,0,...0,1,0.....,0)$

• The set $\{ e_1,e_2,....,e_n\}$ is a basis of $\R^n$ called the standard basis.
• Hence the dimension of $\R^n$ is $n$.

Example:

Example: In terms of Matrices

Rank of a Matrix

Let $A$ be an $m\times n$ matrix.