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Video: Geometric Interpretation of Linear Transformations

Rhodri Jones

Consider the transformation represented by the matrix 3, 0; 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 three zero, zero three. What is the image of the square with vertices zero zero, zero one, one zero, and one one under this transformation? Secondly, what geometric transformation does this matrix represent?

In order to work out the image of the square, we need to multiply each of the vertices by the matrix three zero, zero three. Three zero, zero three multiplied by zero, zero is equal to zero, zero. Therefore, the first point of the image has coordinates zero, zero.

In the same way, multiplying the matrix by the point zero, one gives us new coordinates zero, three. The third vertex becomes three, zero. And finally, the vertex one, one becomes three, three. The four vertices of the image are zero zero, zero three, three zero, and three three.

As each of our coordinates has been multiplied by three, we can say that the geometric transformation that the matrix represents is a dilation or enlargement with scale factor three and centre the origin.