Thursday continue isomorphisms Two vector spaces over same F; T isomorphism between V and W if it is invertible Argument that shows that S T = I v of V to V is identical to T S = I w of W to W in finite VS; major proof Application to Matrices argue by using concepts, vector spaces, linear maps, not matrices makes certain things clear; use insights with concrete statements about numbers - Numerical representation - can use columns of numbers and arrays of columns Major development of Q, Q inverse establish isomorphism; equivalent to choice of basis how do they change from basis to basis Invertible Maps Skip 3E. From 3D to 3F Midterm in two weeks; up to next Thursday post mock exam next week Tu 27 is review; Th 29 is exam; ## Dual Space -