Question

Consider the circular face in the accompanying figure. For each of the matrices A in Exercises 24 through 30, draw a sketch showing the effect of the linear transformation T() = Ax on this face. 1

Answer 1

The transformation T(x) is achieved by rotating the vector x counterclockwise by an angle of 90 degrees.

Let the matrix be,

A =
`\left[\begin{array}{ccc}0&-1\\1&0\end{array}\right]`

Then,

Ax = Ax Tx
`\left[\begin{array}{ccc}0&-1\\1&0\end{array}\right]` x

The matrix is

Tx =
`\left[\begin{array}{ccc}0&-1\\1&0\end{array}\right]`
`\left[\begin{array}{ccc}x1\\x2\end{array}\right]`Tx =
`\left[\begin{array}{ccc}-x2\\x1\end{array}\right]`

The graph is in the image attached below

The transformation T(x) is achieved by rotating the vector x counterclockwise by an angle of 90 degrees.

A mathematical matrix is a rectangular array of numbers or symbols arranged in rows and columns. Matrices are used to represent linear transformations of objects or to solve systems of linear equations. The size of a matrix is determined by the number of rows and columns it contains. Matrices can be added, subtracted, multiplied, and transformed using various operations. They are widely used in fields such as physics, engineering, computer science, and economics for modeling and analyzing complex systems.