How to Determine Whether a Function is a Linear Transformation
Any function that deals with vectors is a transformation, but not every such function is a linear transformation. For it to be linear, the function or transformation of the sum of any two vectors A and B that are part of the domain must be equal to the transformation of vector A plus the transformation of vector B, or T(A+B) = T(A) + T(B). In addition, T(cA), where c is any arbitrary scalar constant, must be equal to c multiplied by the transformation of A, or T(cA) = cT(A).
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Instructions
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Add the two vectors into a new vector V. Given the function T(x1, x2) = (x1 + x2, 3x1) with the 2-tuple vectors A (a1, a2) and B (b1, b2), add vector A to vector B to get the following 2-tuple vector V: (a1 + b1, a2 + b2).
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Take the transformation of vector V. Given the function T(x1, x2) = (x1 + x2, 3x1) and the vector V (a1 + b1, a2 + b2), solve for T(V): (a1 + b1 + a2 + b2, 3(a1 + b1)), or (a1 + a2 + b1 + b2, 3a1 + 3b1).
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3
Take the transformation of the two original vectors. Given the function T(x1, x2) = (x1 + x2, 3x1) and the 2-tuple vector A (a1, a2), take its transformation by solving T(a1, a2): (a1 + a2, 3 * a1) or (a1 + a2, 3a1). Likewise, take the transformation of vector B by solving T(b1, b2): (b1 + b2, 3 * b1) or (b1 + b2, 3b1).
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4
Add the transformations of the two vectors and compare the resulting vector to vector V. Given T(A) = (a1 + a2, 3a1) and T(B) = (b1 + b2, 3b1), add the two vectors to get (a1 + a2 + b1 + b2, 3a1 + 3b1). Compare this to vector V, which is (a1 + a2 + b1 + b2, 3a1 + 3b1). Since they are both the same, this function has passed one of the criteria regarding linear transformations.
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5
Choose a vector, multiply it by an arbitrary scalar constant and then take its transformation to get vector K. Given the function T(x1, x2) = (x1 + x2, 3x1) and vector A (a1, a2), multiply vector A by the arbitrary constant 5 to get A (5a1, 5a2). Then solve for T(5a1, 5a2): (5a1 + 5a2, 15a1). Thus vector K is (5a1 + 5a2, 15a1).
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6
Take the transformation of the original vector, multiply it by the arbitrary scalar constant and compare the resulting vector to vector K. Given the example, multiply the transformation of vector A (a1 + a2, 3a1) by the constant 5 to get (5a1 + 5a2, 15a1). Compare this to vector K (5a1 + 5a2, 15a1). They are both equal, which means the function T(x1, x2) = (x1 + x2, 3x1) has passed the second test. Therefore, it is a linear transformation.
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Tips & Warnings
The two vectors you choose should have the same number of tuples as the function's domain.
