HW$5.$ Kernel $(20$ points$)$ Your Name: $$

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$1.(10$ points$)$ Please briefly describe what you understand about Kernel. Answer the questions such as:

$1)$ Why do you use mapping function in SVMs$?$

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$2)$ What is a kernel function?

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$3)$ What other kernel functions do you know other than polynomial kernel?

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$2.(5)$ Please prove that the function K$($x$1,$ x$2)=(1+$ x$1$ x$2)6$ is a kernel function which can map data x$1$ and x$2$ in R$1$ to R$7$using the mapping function:

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$3.(5)$ In RBF SVMs model, what are going to change if you tune the parameters $($C$,$ $)$ separately?