\(m_{11}x+m_{12}y+m_{13}z=v_1\)
\(m_{21}x+m_{22}y+m_{23}z=v_2\)
\(m_{31}x+m_{32}y+m_{33}z=v_3\)
We can write the above as:
\(\mathbf{M}x=\mathbf{v}\)
What are the properties of \(\mathbf{M}\) and \(\mathbf{v}\)?
They are linear in addition and scalar multiplication.
The rank of a matrix is the dimension of the span of its component columns.
\(rank (M)=span(m_1,m_2,...,m_n)\)
The span of the rows is the same as the span of the columns.
A matrix where every element is \(0\). There is one for each dimension of matrix.
\(A=\begin{bmatrix}0& 0&...&0\\0 & 0&...&0\\...&...&...&...\\0&0&...&0\end{bmatrix}\)
A matrix where \(a_{ij}=0\) where \(i < j\) is upper triangular.
A matrix where \(a_{ij}=0\) where \(i > j\) is lower triangular.
A matrix which is either upper or lower triangular is a triangular matrix.
All symmetric matrices are square.
The identity matrix is an example.
A matrix where \(a_{ij}=a_{ji}\) is symmetric.
A matrix where \(a_ij=0\) where \(i\ne j\) is diagonal.
All diagonal matrices are symmetric.
The identity matrix is an example.
For any two bases, there is a unique linear mapping from of the element vectors to the other.