In support vector machines, I understand it would be computationally prohibitive to calculate a basis function at every point in the data set. However, it is possible to find this optimal solution due to the so-called kernel trick.
Other answers to this question use advanced math and statistics jargon to answer the question (I assume) properly, causing it to be inaccessible to general a data science audience. Could someone post a "big-picture" description (i.e., not necessarily thorough or technically complete) illustrating what the kernel trick is and how it works?