Determine if Linear

Problem

T*([a],[b],[d])=[[2*a−b+5*d],[−4*a+2*b−10*d]]

Solution

1. Identify the standard form of a linear transformation. A transformation T:ℝn→ℝm is linear if it can be written in the form T(v)=A*v for some m×n matrix A

2. Rewrite the transformation by factoring out the variables a b and d from the output vector.

T*([a],[b],[d])=a*[[2],[−4]]+b*[[−1],[2]]+d*[[5],[−10]]

3. Construct the standard matrix A using the coefficients of the variables.

A=[[2,−1,5],[−4,2,−10]]

4. Verify the properties of linearity. Since the transformation is defined by T(x)=A*x where each component is a homogeneous linear combination of the input variables (no constant terms or non-linear powers), it satisfies T*(u+v)=T(u)+T(v) and T*(c*u)=c*T(u)

Final Answer

T*([a],[b],[d])=[[2*a−b+5*d],[−4*a+2*b−10*d]]* is linear

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