Linear Transformation


Linear transformations. Ldo, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

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A linear transformation is a special case of a vector transformation with two properties: addition must be preserved, and scalar multiplication must be preserved.

How to check whether your transformation is linear.

  1. Addition must be Preserved: To check if addition is preserved, take two vectors u and v and add them together. Then transform each vector individually and add the results. If the sum of the transformed vectors is the same as the sum of the original vectors, then addition is preserved and your transformation is linear.
  2. Scalar Multiplication must be Preserved: To check if scalar multiplication is preserved, take a vector u and multiply it by a scalar c. Then transform u and multiply the result by scalar c. If the product of the transformed vector is the same as the product of the original vector, then scalar multiplication is preserved by your transformation and it is linear.

Linear Transformation Example

Example Question: Is the following transformation a linear transformation?
T(x,y)→ (x – y, x + y, 9x)

Step 1: Give the vectors and (from rule 1) some components. I’m going to use and here, but the choice is arbitrary:

  • = (a1, a2)
  • = (b1, b2)
matrix addition linear transformation

Step 2: Find an expression for the addition part of the left side of the Rule 1 equation (we’re going to do the transformation in the next step):
(u + v) = (a1, a2) + (b1, b2)
Adding these two vectors together, we get:
((a1 + b1), (a2 + b2))
In matrix form, the addition is:

Step 3: Apply the transformation. We’re given the rule T(x,y)→ (x – y, x + y, 9x), so transforming our additive vector from Step 2, we get:

  • T ((a1 + b1), (a2+ b2)) =
  • ((a1 + b1) – (a2 + b2),
  • (a1 + b1) + (a2 + b2),
  • 9(a1 + b1)).

Simplifying/Distributing using algebra:
(a1 + b1 – a2 – b2,
a1 + b1 + a2 + b2,
9a1 + 9b1).
Set this aside for a moment: we’re going to compare this result to the result from the right hand side of the equation in a later step.

Step 4: Find an expression for the right side of the Rule 1 equation, T(u) + T(v). Using the same a/b variables we used in Steps 1 to 3, we get:
T((a1,a2) + T(b1,b2))

Step 5: Transform the vector u, (a1,a2). We’re given the rule T(x,y)→ (x – y, x + y, 9x), so transforming vector u, we get:

  • (a1 – a2,
  • a1 + a2,
  • 9a1)

Step 6: Transform the vector v. We’re given the rule T(x,y)→ (x – y, x + y,9x), so transforming vector v, (a1,a2), we get:

  • (b1 – b2,
  • b1 + b2,
  • 9b1)

Step 7: Add the two vectors from Steps 5 and 6:
(a1 – a2, a1 + a2, 9a1) + (b1 – b2, b1 + b2, 9b1) =
((a1 – a2 + b1 – b2,
a1 + a2 + b1 – b2,
9a1 + 9b1)

Step 8: Compare Step 3 to Step 7. They are the same, so condition 1 (the additive condition) is satisfied.

Part Two: Is Scalar Multiplication Preserved?

In other words, in this part we want to know if T(cu)=cT(u) is true for T(x,y)→ (x-y,x+y,9x). We’re going to use the same vector from Part 1, which is = (a1, a2).

Step 1: Work the left side of the equation, T(cu). First, multiply the vector by a scalar, c.
c * (a1, a2) = (c(a1), c(a2))

Step 2: Transform Step 1, using the rule T(x,y)→ (x-y,x+y,9x):
(ca1 – ca2,
ca1 + ca2,
9ca1)
Put this aside for a moment. We’ll be comparing it to the right side in a later step.

Step 3: Transform the vector u using the rule T(x,y)→ (x-y,x+y,9x). We’re working the right side of the rule 2 equation here:
(T(a1, a2)=
a1 – a2
a1 + a2
9a1)

Step 4: Multiply Step 3 by the scalar, c.
(c(a1 – a2)
c(a1 + a2)
c(9a1))
Distributing c using algebra, we get:
(ca1 – ca2,
ca1 + ca2,
9ca1)

Step 5: Compare Steps 2 and 4. they are the same, so the second rule is true. This function is a linear transformation.

Conclusion

In order to check if your transformation is linear, you must first determine if addition is preserved under your transformation. To do this, add two vectors together and then transform each vector individually before adding the results. If the sum of the transformed vectors equals the sum of the original vectors, then addition is said to be “preserved” and your transformation passes this test.
You must also determine if scalar multiplication is preserved under your transformation. To do this, multiply a vector by a scalar and then transform that vector before multiplying the result by your scalar again. If the product of the transformed vector equals the product of multiplying your original vector by your scalar, then scalar multiplication has been “preserved” and your transformation passes this test— congratulations! Your transformation is linear!


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