Find the Nullity f(t)=[[cos(2t)],[sin(t)]]

Problem

ƒ(t)=[[cos(2*t)],[sin(t)]]

Solution

1. Identify the nature of the function. The function ƒ(t) is a vector-valued function mapping a scalar t∈ℝ to a vector in ℝ2

2. Define the nullity in the context of a linear transformation. The nullity is the dimension of the kernel (null space). For a function to have a nullity in the linear algebraic sense, it must be a linear transformation.

3. Check for linearity. A function L is linear if L*(u+v)=L(u)+L(v) and L*(c*u)=c*L(u) For ƒ(t) we check ƒ(0)

ƒ(0)=[[cos(0)],[sin(0)]]=[[1],[0]]

4. Determine if the function is linear. Since ƒ(0)≠0 the function ƒ(t) is not a linear transformation.

5. Conclude the result. Because ƒ(t) is not a linear transformation, the concept of "nullity" (the dimension of the kernel of a linear map) does not apply to this function in standard linear algebra.

Final Answer

Nullity of *ƒ(t)* is undefined because *ƒ(t)* is not a linear transformation.

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