Let U be an m m matrix, let V be an n n matrix, and let = 1 O where 1 is an n n diagonal matrix with diagonal entries 1, 2, ... , n and O is the (m n) n zero matrix. (a) Show that if U = (U1, U2), where U1 has n columns, then U = U11 (b) Show that if A = UVT , then A can be expressed as an outer product expansion of the form A = 1u1vT 1 + 2u2vT 2 ++ nunvT n

Consciousness and Sleep What is consciousness ● wakefulness ○ the difference between being asleep and being awake ● voluntary action ○ deliberate control of behavior ● awareness ○ focus of attention ○ accessing own internal systems - you can think about yourself ○ reportability of mental states ○ perception of an “I” who observes, acts, “runs the system” I. Sleep A. why do we sleep 1. humans spend ⅓ of their life asleep even though you can’t mate, eat, or protect yourself when you’re asleep - has big evolutionary costs B. environmental pressures 1. different environments have different adaptive problems and