*Gerald Lemay*Show bio

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

Instructor:
*Gerald Lemay*
Show bio

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.

Go north 2.5 miles, turn left â€¦ 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**.

Look at *y* = *x* and *y* = *x*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* = *x*2 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 *u*1 and *u*2. These will be mapped from domain to range (represented by *v*) with a transformation T. In other words, T maps *u*1 into *v*1 and T maps *u*2 into *v*2.

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 Î±.

Selecting values makes the properties clearer. What if *u*1 is 2 and *u*2 is 3? Using *y* = *x*:

Transforming 2 using *y* = *x* produces a range value *v*1 equal to 2. Same idea for a domain value of 3: the resulting *v*2 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 *v*1 and the *v*2.

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(Î±*u*) = Î±T(*u*), try plugging in some values for Î± and *u*.

For example, if *u* = 2 then it gets transformed to a range value T(*u*) = 2. If Î± is 3, then Î±T(*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.

Looking at Î±*u* 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* = *x*2 ? Time to choose values in the domain. Let *u*1=2 and *u*2 = 3. Add and then transform.

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

4 + 9 is 13 which is clearly not 25. Transforming after adding is not the same as adding after transforming: the first property is not satisfied. We can stop checking and conclude that *y* = *x*2 is not a linear transformation. But just for practice, what about the second property? Does T(Î±*u*) equal Î±T(*u*) ? Again, with numbers like *u* = 2 and Î± = 3, we are checking if multiplying by a constant after transforming is the same as transforming after multiplying.

Not the same. The second property is not satisfied. Same conclusion: this transformation is not linear.

We know that the transformation *y* = *x* + 1 is a straight line.

Here's why this transformation is not linear. Choosing some values: *u*1 = 2 and *u*2 = 3. Then, *u*1 + *u*2 = 5, T(*u*1) = 3, T(*u*2) = 4, T(*u*1 + *u*2) = 6.

Does T(*u*1 + *u*2) = T(*u*1) + T(*u*2) ? This is asking if 6 is equal 3 + 4. Not equal. Thus, this transformation is not linear. The second property won't be satisfied either.

It also turns out a linear transformation always maps 0 to 0. At the **origin**, *x* = 0 and *y* = 0. A line can only possibly be a linear transformation if the line passes through the origin.

Now we know more about getting from 'here' to 'there'!

**Transformations** map numbers from **domain** to **range**. If a transformation satisfies two **defining properties**, it is a **linear transformation**. The **first property** deals with addition. It checks that the transformation of a sum is the sum of transformations. The **second property** deals with multiplying by a constant, to check if multiplying after transforming is the same as transforming after multiplying.

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