Question Video: Geometric Interpretation of Linear Transformations Mathematics

Consider the transformation represented by the matrix [−3, 0 and 0, −3]. What is the image of the square with vertices (0, 0), (0, 1), (1, 0), and (1, 1) under this transformation? What geometric transformation does this matrix represent?

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Video Transcript

Consider the transformation represented by the matrix negative three, zero, zero, negative three. What is the image of the square with vertices zero, zero; zero, one; one, zero; and one, one under this transformation? And what geometric transformation does this matrix represent?

The first thing we wanna do is take the transformation matrix we were given and use it to find the image of the square. To do that, we take the transformation matrix and we multiply it by the first vertex of the square zero, zero. This multiplication will still give us the coordinate zero, zero. We follow the same procedure for the second vertex. This time, we get a coordinate of zero, negative three, then the third vertex which gives us the point negative three, zero. And the final vertex under this transformation is negative three, negative three.

At this point particularly, if you haven’t worked with very many of these type of transformations, it might be hard for you to imagine what this looks like. And if that’s the case, you could sketch the square and its image. We’ll sketch the original square in pink — zero, zero; zero, one; one, zero; and one, one. And its image we can sketch in blue — zero, zero; zero, negative three; negative three, zero; and negative three, negative three. We can say that our original square had a side length of one unit, and the image has a side length of three units.

The first question is asking, what has happened to the original square after its transformation? It is still a square. However, the new square has vertices zero, zero; zero, negative three; negative three, zero; and negative three, negative three.

The second part of our question asks, what geometric transformation does this matrix represent? To answer this, we need to think about what types of transformations there are. One type is a translation. A geometric translation is when a shape is moved without being rotated or changing size. You might also hear this referred to as a slide. Another type of geometric transformation is a reflection. In a reflection, every point is the same distance from the central line, and the image has the same shape and size as the original.

A third type of geometric transformation is a dilation. A dilation produces an image that is the same shape as the original, but a different size. A dilation has two components: a scale factor or ratio and a center. Because we know that the translation and reflection both require the image to be the same size as the original, we know that our transformation is not a translation or a reflection. So we can say this transformation is a dilation.

It also means we need to identify the scale factor and the center. We do recognize that our new square is three times the size of our old square. However, because our image has been rotated 180 degrees about the origin from the original, its scale factor will be negative. The integer three tells us how many times larger the image is from the original. And the negative value tells us that our image was rotated 180 degrees from the original.

When we look at our sketch, we can see that the center of this transformation is at the origin. And so we can say that the matrix negative three, zero, zero, negative three represents a dilation with a scale factor of negative three and center at the origin.

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