2. Do the DCT on several vectors and explain result.

The first vector is {3,3,3,3}, which is just a multiple of the first row.  The DCT will have zero amounts of the other three.

dct . {3, 3, 3, 3}

RowBox[{{, RowBox[{12., ,, 2.22045*10^-16, ,, 0., ,, RowBox[{-, 1.05471*10^-15}]}], }}]

The second vector is {-10,10,-10,10}.  The DCT is mostly high frequency.

dct . {-10, 10, -10, 10}

RowBox[{{, RowBox[{0., ,, RowBox[{-, 10.8239}], ,, 0., ,, RowBox[{-, 26.1313}]}], }}]

The third vector is {1,2,3,4}.  This has a strong constant offset, then a fair amount of the next basis, and little of high frequencies.

dct . {1, 2, 3, 4}

RowBox[{{, RowBox[{10., ,, RowBox[{-, 3.15432}], ,, 0., ,, RowBox[{-, 0.224171}]}], }}]

Created by Mathematica  (April 1, 2004)