Gradient of a linear function

L. El Ghaoui

Gradient of a linear function

Consider the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ , with values

$f(x) = x_1 + 2 x_2.$

Its gradient is constant, with values

$\nabla f=\left(\begin{array}{c} \frac{\partial f}{\partial x_1}(x) \\ \frac{\partial f}{\partial x_2}(x) \end{array}\right)=\left(\begin{array}{c} 1 \\ 2 \end{array}\right).$

For a given $t \in \mathbb{R}$ , the -level set is the set of points such that $f(x)=t$ :

${\bf L}_t(f):=\{(x_1, x_2): x_1 + 2x_2 = t\}$

The level sets are hyperplanes and are orthogonal to the gradient.

Gradient of a linear function

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