P15) Finite, sampled signals and their discrete Fourier transforms both live in — a *vector space. *Show that the DFT respects basic vector space operations — addition and scalar multiplication. More precisely, show that (a) if and are signals in and , then , and (b) if , is a scalar, and , then .

P16) Let and . Calculate and .

P17) Let and let be the matrix whose -th entry is , . Show that (a) is a symmetric matrix; (b) , where is the matrix obtained by conjugating each entry of ; (c) if we view as a column vector, then .

P18) Write down the matrix of the previous problem in the case . (Each entry should be expressed in standard for .) Use it to compute and , where and .

P19) Make some stem plots of the modulus of , where where the string of ones has length and the string of zeroes has length . (We computed in class, but you should check it with MATLAB.)