Linear Transformations: Properties & Examples

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

A mapping from one space to another is called a transformation. A transformation is linear if two properties are satisfied. In this lesson we use these properties to identify linear transformations.

Linear Transformations

Go north 2.5 miles, turn left … These might be directions for getting from here to there. In math, 'here' and 'there' can correspond to 'domain' and 'range'. The domain consists of the values on the x-axis, while the range consists of the values on the y-axis. Think of a function y = f(x) as a set of directions: a transformation of x values into y values. For example, the transformation y = x + 2 says that to transform the x-values into y-values, take the x values and add 2. Specifically, for a domain value of x = 1, the transformation x + 2 leads to a range value y = 1 + 2 = 3. The term map is also used for this action of getting from 'here' to 'there'. In short, a transformation maps values from the domain to values in the range.

In this lesson, we'll study a special type of transformation called the linear transformation.

Defining the Linear Transformation

Look at y = x and y = x2.

y = x
y=x

y = x 2
y=x%%(vertical-align:super)2%%.


The plot of y = x is a straight line. The words 'straight line' and 'linear' make it tempting to conclude that y = x is a linear transformation. The other plot is clearly not a straight line; our intuition says y = x2 is not a linear transformation. You know what? For these examples, our intuition is correct!

But as will be demonstrated later in this lesson, intuition can lead to errors. We need a more solid way to define a linear transformation.

Let's look at two domain values u1 and u2. These will be mapped from domain to range (represented by v) with a transformation T. In other words, T maps u1 into v1 and T maps u2 into v2.

We write the mapping as:

Now here's the solid way to check if a transformation is linear. A transformation T is a linear transformation if both of the following properties are true:

and

We can choose any number for the constant α.

Testing With Values

Selecting values makes the properties clearer. What if u1 is 2 and u2 is 3? Using y = x:

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Transforming 2 using y = x produces a range value v1 equal to 2. Same idea for a domain value of 3: the resulting v2 is 3. Now if we add the two domain values, 2 + 3, we get 5. Transforming this sum of 5 produces a range value of 5. This value is the same as the sum of v1 and the v2.

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That's what we are checking: can we transform the sum of two domain values and get the same sum when these values are transformed separately? For y = x, this works. Great! The first property is satisfied. This may have seemed obvious but y = x is a pretty simple transformation.

To test the second property, T(αu) = αT(u), try plugging in some values for α and u.

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For example, if u = 2 then it gets transformed to a range value T(u) = 2. If α is 3, then αT(u) is 3 times 2, which is 6. The property we are checking for is whether multiplying a domain value by a number, and then transforming, gives the same result as multiplying after transforming.

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Looking at αu we get 3 times 2 which is 6. Transforming 6 into the range, we get 6. The second property is satisfied. Conclusion: the transformation y = x is a linear transformation!

What about y = x2 ? Time to choose values in the domain. Let u1=2 and u2 = 3. Add and then transform.

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Does this equal the sum of those domain values transformed separately?

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