Rank

68. Rank

Dimension of the column space

rank (𝐴) = dim (𝐶 (𝐴))

\begin{matrix} 𝑎_{11} 𝑥_{1} + 𝑎_{12} 𝑥_{2} + \dots + 𝑎_{1 𝑚} 𝑥_{𝑚} = 𝑏_{1} \\ 𝑎_{21} 𝑥_{1} + 𝑎_{22} 𝑥_{2} + \dots + 𝑎_{2 𝑚} 𝑥_{𝑚} = 𝑏_{2} \\ ⋮ \\ 𝑎_{𝑛 1} 𝑥_{1} + 𝑎_{𝑛 2} 𝑥_{2} + \dots + 𝑎_{𝑛 𝑚} 𝑥_{𝑚} = 𝑏_{𝑛} \end{matrix}

Coefficient Matrix ( $𝐴$ ):

𝐴 = [\begin{matrix} 𝑎_{11} & 𝑎_{12} & \dots & 𝑎_{1 𝑚} \\ 𝑎_{21} & 𝑎_{22} & \dots & 𝑎_{2 𝑚} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 𝑎_{𝑛 1} & 𝑎_{𝑛 2} & \dots & 𝑎_{𝑛 𝑚} \end{matrix}]

Variable Vector (x):

x = [\begin{matrix} 𝑥_{1} \\ 𝑥_{2} \\ ⋮ \\ 𝑥_{𝑚} \end{matrix}]

Constant Vector (b):

b = [\begin{matrix} 𝑏_{1} \\ 𝑏_{2} \\ ⋮ \\ 𝑏_{𝑛} \end{matrix}]

Ax = 𝑏

Example

The system of equations:

\begin{matrix} 2 𝑥_{1} + 3 𝑥_{2} + 5 𝑥_{3} = 100 \\ 4 𝑥_{1} + 2 𝑥_{2} + 1 𝑥_{3} = 80 \\ 1 𝑥_{1} + 5 𝑥_{2} + 2 𝑥_{3} = 60 \end{matrix}

Can be represented as a matrix equation:

[\begin{matrix} 2 & 3 & 5 \\ 4 & 2 & 1 \\ 1 & 5 & 2 \end{matrix}] [\begin{matrix} 𝑥_{1} \\ 𝑥_{2} \\ 𝑥_{3} \end{matrix}] = [\begin{matrix} 100 \\ 80 \\ 60 \end{matrix}]