MATH-3110 Lecture 1: 1.3 vector arithmetic

Consider F and C below. F(x, y)i 2y arctan(x) j 1 + X c: r(t) = t2 i + 3t j, 0 t 1 (a) Find a function f such that F = Vf f(x, y) = |y?arctan(x) 2 (b) Use part (a) to evaluate V. dr along the given curve C.

1. (100%) Let T : Rn-> Rrn be a one-to-one linear transformation and V be a subspace of Rn. (a) (b) (30%) Show that W-(T(u) : u E V} is a subspace of R". (40%) Prove that if {ui,Uz, W. , uk} įs a basis for V, {T(u), T(u2), ,T(w)} is a basis for (c) (30%) Prove that dim V-dim W