Teaching CSIR NET Mock Test Series Mathematical Science Analysis Derivative as a Linear Transformation
Let \(f: \mathbb{R}^n \rightarrow \mathbb{R}\) be the map \(f\left(x_1, \ldots, x_n\right)=\) \(a_1 x_1+\cdots+a_n x_n\) where a = (a1,..., an) is a fixed non zero vector. Let Df(0) denote the derivative of f at 0.
Which of the following is/are true?
1
(Df)(0) is a linear map from \(\mathbb{R}^n \rightarrow \mathbb{R}\)
2
\([(D f)(0)](a)=\|a\|^2\)
3
\([(D f)(0)](a)=0\)
4
\([(D f)(0)](b)=a_1 b_1+\cdots+a_n b_n\), for \(b=\left(b_1, \ldots, b_n\right)\)