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Matrices as Linear Transformations – I

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Introduction


A Matrix is a collection of numbers represented as a rectangular array/ list.
2D Matrices are distinguished using the number of rows and columns.
Example:

    \[A = \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\end{bmatrix}\]


The elements of A can be denoted using A_{ij}

The order of a matrix is defined as a tuple denoting the number of rows and columns that the matrix has. It can also be seen as the dimensions of a matrix. For instance, the above matrix has order (2, 3).

Linear Transformation

The multiplication of a vector by a matrix “transforms” the input vector into an output vector. After this transformation, the size of the resultant vector may or may not be the same as that of the input vector. Such a matrix multiplication which happens in a linear fashion is termed as a linear transformation.

Linear Transformation can be seen as a transformation of complete space on its own but is a conditional one.
A Linear transformation in any dimension can be expressed in the form of Matrix-vector multiplication.
                                                             A * \vec{v} = \vec{v'}

There are a few properties that every transformation must have to be called as a Linear transformation:
1. All lines must remain lines after the transformation (should not curve).
2. The origin should always remain the same.

Examples
1. Rotation:

2. Shear:

Operations on matrices

    1. Matrix-Matrix addition:

      The addition of two matrices is done element-wise. For addition to be done, the order of both the matrices needs to be identical.
      The geometrical significance of the addition of two matrices can be considered as computing the sum of two linear transformations.

      Example:               \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix}\begin{bmatrix}9 & 8 & 7\\6 & 5 & 4\\3 & 2 & 1\end{bmatrix} = \begin{bmatrix}10 & 10 & 10\\10 & 10 & 10\\10 & 10 & 10 \end{bmatrix}

    2. Matrix-Matrix Multiplication:

      Similar to a composition of two functions on a variable, the composition of two matrices can be considered as performing two linear transformations on a vector or space.
                                                 A(B * \vec{v}) = A * \vec{v'} = \vec{v''}

      We can only multiply two matrices if their dimensions are compatible, which means the number of columns in the first matrix is the same as the number of rows in the second matrix.

      If A = [a_{ij}] is an m×n matrix and B = [b_{ij}] is an n×p matrix, the product AB is an m×p matrix.
      AB=[c_{ij}] , where c_{ij} = a_{i1} * b_{1j} + a_{i2} * b_{2j} + ... + a_{in} * b_{nj}.


      Example:                 \begin{bmatrix}2 & 7 & 3\\1 & 5 & 8\\ 0 & 4 & 1\end{bmatrix}\begin{bmatrix}3 & 0 & 1\\2 & 1 & 0\\1 & 2 & 4\end{bmatrix} = \begin{bmatrix}23 & 13 & 14\\21 & 21 & 33\\9 & 6 & 4 \end{bmatrix}

    3. Transpose of a Matrix:

      Transpose of a matrix is obtained by changing the rows and columns of a matrix.
      The transpose of a given matrix A is represented as A^T

      Example:              \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix}^T\begin{bmatrix}1 & 4 & 7\\2 & 5 & 8\\3 & 6 & 9\end{bmatrix}

    4. Dot Product of Matrices:

      The dot product between two matrices is column-wise multiplication of corresponding elements.
      Geometrically, \vec{a} \cdot \vec{b} is the length of vector \vec{a} multiplied length of \vec{b} and cosine of angle between them.

      Example:     Let \vec{a} = \begin{bmatrix} a_1 & a_2 \end{bmatrix} , \vec{b} = \begin{bmatrix} b_1 & b_2 \end{bmatrix}
                                    \vec{a} \cdot \vec{b} = a_1*b_1 + a_2*b_2
                                    a_1*b_1 + a_2*b_2 = ||a|| * ||b|| * \cos \theta

Properties of matrices

  1. Commutativity:
    If M_1, M_2 are matrices, of such order that these operations can be applied,
                              M_1 + M_2 = M_2 + M_1
                              M_1 \cdot M_2 \neq M_2 \cdot M_1

  2. Associativity:
    If A, B, C are matrices, of such order that these operations can be applied,
                              A + (B + C) = (A + B) + C
                              A \cdot (B \cdot C) = (A \cdot B) \cdot C
  3. Distributive:
    If A, B, C are matrices, of such order that these operations can be applied,
                              A \cdot (B + C) = A \cdot B + B \cdot C
                              (A + B) \cdot C =A \cdot C + B \cdot C

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