Dashboard

Now you are in the subtree of Container for Linear Algebra project. 

The set of m by n matrices is a vector space.

Created over 8 years ago, updated 10 days ago

Theorem: The set (M_{m imes n}(\mathbb{F})) of all (m imes n) matrices with entries in (\mathbb{F}) is a vector space over (\mathbb{F}).

  • Addition: ((A + B){ij} = A{ij} + B_{ij}) (entrywise)
  • Scalar multiplication: ((cA){ij} = c \cdot A{ij})

Zero vector: The zero matrix (all entries are 0).

Dimension: (\dim(M_{m imes n}) = mn).

Standard basis: The matrices (E_{ij}) with a 1 in position ((i,j)) and 0 elsewhere. There are (mn) such matrices.

Example: (M_{2 imes 2}(\mathbb{R})) has dimension 4 with basis:
[egin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix}, egin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}, egin{pmatrix} 0 & 0 \ 1 & 0 \end{pmatrix}, egin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix}]