# In this example, which of these vectors are support vectors?

The hyperplane of hard margin SVM with $$\phi$$ kernel is calculated as following that input space using $$\phi$$ to map to higher dimension space.

$$f(\phi(x))=4\phi_1(x)+9\phi_2(x)+4\phi_3(x)$$

$$\phi(x)=\left[ \begin{array}{c} \phi_1(x)=x_1^2\\ \phi_2(x)=x_2^2\\\phi_3(x)=x_1x_2\\\phi_4(x)=-x_1 \end{array} \right]$$

Which of the following can be a support vector?

$$x=\left[ \begin{array}{c} +1\\ 1 \end{array} \right]$$ $$y=\left[ \begin{array}{c} 1\\ -1 \end{array} \right]$$

Options:

1. $$\ x=$$ YES, $$\ y=$$ No

2. $$\ x=$$ YES, $$\ y=$$ YES

The correct answer is option $$1$$. My challenge is where is the trick in this question, because calculation by hand shown that none of them are support vectors? How can we choose an option here?

• Can you also share how your thinking process is? i.e. why do you think that the given vectors are not SVs? Jan 13 at 17:14
• Please add the self-study tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. Jan 14 at 12:52

the wikipedia entry for SVM is the formula $$w^T x - b$$, (or better $$w^T \phi(x) - b$$ after you use the kernel trick) then points in the support vector should have $$f(\phi(x)) = \pm 1$$ (see this figure in the wiki page).
So just plug in the values of the two points and check whether $$f(\phi(x))$$ and $$f(\phi(y))$$ is either +1 or -1.