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  • Linear Algebra
  • Matrix Space
  • Dim./Basis
  • Subspaces Ops.
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    Dimension and Basis

    Basis of all
    \(3\times 3\)
    matrices

    Our matrix space is something like,
    \[ \begin{bmatrix} \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix} \]
    We have
    \(9\)
    independent entities for every
    \(3\times 3\)
    matrices.
    So there are
    \(9\)
    basis,
    \[ \begin{bmatrix} \color{red}{1} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & \color{red}{1} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & \color{red}{1} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \]
    \[ \begin{bmatrix} 0 & 0 & 0 \\ \color{red}{1} & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ 0 & \color{red}{1} & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & \color{red}{1} \\ 0 & 0 & 0 \\ \end{bmatrix} \]
    \[ \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ \color{red}{1} & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & \color{red}{1} & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \color{red}{1} \\ \end{bmatrix} \]
    We can get any
    \(3\times 3\)
    matrix by the linear combination of these
    \(9\)
    matrices

    What is the Dimension of all
    \(3\times 3\)
    matrix?
    There are
    \(9\)
    basis and by there linear combination we can get all other matrices.
    So Dimension of all
    \(3\times 3\)
    matrices is
    \(9\)
    .

    Basis of all
    \(3\times 3\)
    Lower Triangular matrices

    Our matrix space is something like,
    \[ \begin{bmatrix} \mathbb{R} & 0 & 0 \\ \mathbb{R} & \mathbb{R} & 0 \\ \mathbb{R} & \mathbb{R} & \mathbb{R} \\ \end{bmatrix} \]
    We have
    \(6\)
    independent entities for a
    \(3\times 3\)
    matrix.
    So there are
    \(6\)
    basis,
    \[ \begin{bmatrix} \color{red}{1} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ \color{red}{1} & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ 0 & \color{red}{1} & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \]
    \[ \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ \color{red}{1} & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & \color{red}{1} & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \color{red}{1} \\ \end{bmatrix} \]
    We can get any Lower Triangular matrix by the linear combination of these matrices

    What is the Dimension of all
    \(3\times 3\)
    Lower Triangular matrix?
    There are
    \(6\)
    basis and by there linear combination we can get all other Lower Triangular matrices.
    So Dimension of all
    \(3\times 3\)
    Lower Triangular matrices is
    \(6\)
    .

    Basis of all
    \(3\times 3\)
    Symmetric matrices

    Our matrix space is something like,
    \[ \begin{bmatrix} \mathbb{R} & \color{red}{\mathbb{R}} & \color{blue}{\mathbb{R}} \\ \color{red}{\mathbb{R}} & \mathbb{R} & \color{green}{\mathbb{R}} \\ \color{blue}{\mathbb{R}} & \color{green}{\mathbb{R}} & \mathbb{R} \\ \end{bmatrix} \]
    We have
    \(6\)
    independent entities for a
    \(3\times 3\)
    matrix.
    So there are
    \(6\)
    basis,
    \[ \begin{bmatrix} \color{red}{1} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ \color{red}{1} & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ 0 & \color{red}{1} & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \]
    \[ \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ \color{red}{1} & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & \color{red}{1} & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \color{red}{1} \\ \end{bmatrix} \]
    We can get any Symmetric matrix by the linear combination of these matrices

    What is the Dimension of all
    \(3\times 3\)
    Symmetric matrix?
    There are
    \(6\)
    basis and by there linear combination we can get all other Symmetric matrices.
    So Dimension of all
    \(3\times 3\)
    Symmetric matrices is
    \(6\)
    .