Question

Determine if the given function is a linear transformation. If so, identify the matrix A such that T(x) = Ax. If not, explain why not

a. T(x1, x2) = (x1 ? x2, x1x2)

b. T(x1, x2) = (x2 sin(?/4), x1 ln(2))

Answer 1

Neither function a: T(x1, x2) = (x1 - x2, x1x2) nor function b: T(x1, x2) = (x2 sin(π/4), x1 ln(2)) is a linear transformation, as they don't satisfy the conditions for linearity due to the inclusion of multiplicative and logarithmic operations.

Step-by-step explanation:

To determine if a given function is a linear transformation, we need to check if it satisfies two conditions: (1) T(u + v) = T(u) + T(v) for any vectors u and v, and (2) T(cu) = cT(u) for any vector u and scalar c. If a function satisfies these conditions, it's a linear transformation, and we can represent it by a matrix A such that T(x) = Ax.

Function a: T(x1, x2) = (x1 - x2, x1x2)

This function is not a linear transformation. This is because the second component of the output, x1x2, is not a linear combination of x1 and x2. Since one of the outputs is the product of x1 and x2, it violates the conditions required for a function to be linear.

Function b: T(x1, x2) = (x2 sin(π/4), x1 ln(2))

This function is also not a linear transformation. While the term x2 sin(π/4) is linear (since sin(π/4) is just a constant), the second term, x1 ln(2), is not linear because the natural logarithm function is not a linear operation. In conclusion, neither function is linear, and we can't associate any matrix A with these transformations.