Transforming distance in miles into distance in feet is a linear tranformation.

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This is a linear transformation because you multiply the distance in miles by 5,280.
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You have the times (in seconds) it took each of five runners to run 100 yards. You decide you are more interested in speed than in time, so you transform your data into yards per second by dividing the distance of 100 yards by the number of seconds it took. Is this a linear transformation?

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This is not a linear transformation because your transformation does not involve multiplying time by a constant and adding a constant. Instead, you are dividing by time.
The figure shows yard per second as a function of seconds it took to run 100 yards.

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The relationship is linear because the points fall on a straight line.
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The relationship is not linear. The function levels off somewhat rather than being a straight line.
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The function decreases quickly at first but decreases slower after that. Therefore the relationship is not linear.