Module Theory. Linear Algebra. Contents Problem Hint. If it is true, prove it. If not, give a counterexample. The answer is true. Recall that the transpose of a matrix is the sum of its diagonal entries.
Use the Cayley-Hamilton theorem Proof. There are many counterexamples. If so, prove it. If not, give a counter example. What is the trace of […]. Two Matrices with the Same Characteristic Polynomial. We will be looking here at some basic examples of using matrices to represent different kinds of transformations of two-dimensional objects. Let's start with simple reflection. A single point A with xy coordinates 3,4. Supposing we wish to find the matrix that represents the reflection of any point x , y in the x -axis.
The transformation involved here is one in which the coordinates of point x , y will be transformed from x , y to x , - y. For this to happen, x does not change, but y must be negated.
We can therefore achieve the required transformation by multiplying y by minus one Because we are dealing with a two-dimensional image, all points in the two-dimensional plane are represented by two variables x and y.
The transformations we apply to a point must set values for x and y independently. We will therefore represent our transformation using the two -by- two matrix M :. The original position of point A relative to the origin is given by a position vector x , y which we can represent using a one -by- two column matrix which we'll call matrix V.
By convention, the point created by a transformation is called the image of the original point, and is often given the same label but with the addition of a tick mark a single apostrophe. Thus A will become A'. The position of point A' relative to the origin after the transformation has been applied is given by the position vector x , - y , which we can represent using a second one -by- two column matrix which we'll call matrix V '.
We can find V ' as follows. When we multiply two matrices, each row in the first matrix is multiplied by each column in the second matrix, so we have:. In order for the vales of x and y to be transformed independently , elements b and c in matrix M must be zero.
We therefore have:. So the values of a and d will be one and minus one respectively. We now have:. Substituting the original coordinates for point A into vector V 1 , we get:. Point A' has xy coordinates 3 , - 4. There are a number of simple transformations involving reflection in the coordinate plane that can be achieved using a transformation matrix like the one we saw above.
Here are a few examples:. These transformations work the same with geometric shapes as they do with a single point. The only difference is that each point in the coordinate plane that defines the shape has a separate vector. Consider the triangle shown below:. Triangle ABC has xy coordinates of: 3, 4 , 5, 1 , 1, 1. Since the points that define the triangle are given by three position vectors, we will require a two -by- three matrix to represent them.
Suppose we want to reflect this triangle in the y -axis. The transformation can be achieved as follows:. Triangle A'B'C' has xy coordinates of: -3, 4 , -5, 1 , -1, 1.
We can apply a linear transformation such as reflection to any two-dimensional figure defined by n points in the coordinate plane using the same two -by- two transformation matrix. A two -by- n matrix is used to hold the position vectors for the figure.
It is somewhat trickier to find the transformation matrix for a point that must be reflected in a line which, although it goes through the origin, is defined by different values of x and y.
The form for the matrix is:. Once you know the tangent, the angle can be found using the arctan function on your calculator this is usually labelled tan If we now substitute actual values into our transformation matrix we get:.
Point A' has xy coordinates Example A square has its vertexes in the following coordinates 1,1 , -1,1 , -1,-1 and 1, More classes on this subject Geometry Transformations: Common types of transformation Geometry Transformations: Vectors.
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