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Algebra of Matrices

4.3 Using Matrices for Transformations and its Inverse

Using Matrices for Transformations

  • Some transformations can be represented as matrices. For instance, all linear transformations can be shown by matrices.

Note: Linear transformation is a general type of transformation that will not be studied in this course. Reflection and dilation are actually particular types of linear transformation, while translation is not.

  • Generally, linear transformations (which includes reflection and translation) can be described as:

\left[\begin{array}{l} x \\y\end{array}\right] \rightarrow\left[\begin{array}{ll}a & b \\c & d\end{array}\right]\left[\begin{array}{l}x \\y\end{array}\right]=\left[\begin{array}{l}a x+b y \\c x+d y\end{array}\right]

which describes the rule (x,\ y) \rightarrow(a x+b y,\ c x+d y).

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4.2 Background: Introduction to Matrices

Introduction to Matrices

  • A matrix is a rectangular array of numbers. The numbers in the array are called the entries of the matrix. The following are examples of matrices:

\left[\begin{array}{cc}-3 & 4 \\5 & 6\end{array}\right] \quad\left[\begin{array}{l}5 \\6 \\7\end{array}\right]\quad\left[\begin{array}{ccc}\sqrt{2} & \pi & 3 \\0 & 0 & 1 \\\sqrt{2} & 0 & \pi\end{array}\right] \quad[5]

  • As you can see above, each matrix have a certain number of rows (horizontal lines) and columns (vertical lines). Together we call it the size, or dimension of a matrix. For the examples above, their sizes (in order) are:

2 x 2, 3 x 1, 3 x 3, 1 x 1

To summarise, we write a matrix’s size as (no. of rows) x (no. of columns).

  • We denote a_{ij} as the entry of a matrix in row i and column j.
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