Gerald has taught engineering, math and science and has a doctorate in electrical engineering.
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.
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:
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:
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.
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.
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.
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.
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 = x2 is not a linear transformation. But just for practice, what about the second property? Does T(αu) equal αT(u) ? Again, with numbers like u = 2 and α = 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.
A Non-Intuitive Example
We know that the transformation y = x + 1 is a straight line.
Here's why this transformation is not linear. Choosing some values: u1 = 2 and u2 = 3. Then, u1 + u2 = 5, T(u1) = 3, T(u2) = 4, T(u1 + u2) = 6.
Does T(u1 + u2) = T(u1) + T(u2) ? 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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