True or FalseIf L : N4 -+ R' is a line;lr transfonnation. then it is possiblethat dim ker L = I and dim range L = 2.

1/29/18 Lecture Monday, January 2912:10 PM Multiplying Vectors: The Dot Product. Challenges: a) Prove that v • w = |v||w|cos(θ) b) Prove the triangle inequality: |a+b|<= |a| + |b| Can we "multiply" v and w Yes. Dot product and cross product. • Def. v=3 w = **3 in Why can we multiply only each part, and not every single component with each of the other Let's try to multiply each component with the next to prove why. For the basis vectors, dotting any basis vector with a different one equals zero! Dotting the same basis vector with itself yields 1. Properties of • product 2 A) v • v = |v| . B) commutativity. u • v = v • u C) distributivity. v • (w+u) = v • w + v • u D) scalar associativity. c(v • u) = cv • w = v • cw E) 0 x v = 0, and c x 0 = 0. Note: there is no multiple dot product property, i.e. v • w • u does not work because evaluating a pair would yield a number, and you cannot dot a number with a vector. Dot product applications v • w captures |v|,|w|, and the angle between the vectors, where the angle returned is between 0 and pi. Theorem. Orthogonal Projections Note: if you don't add the direction, you get the scalar projection: Direction angles and cosines 1/31/17 Lecture Wednesday, January 12:09 PM Challenges: Prove that |v x w| = |v||w|sin(angle between. Prove that v x (w x u) = (v•u)w (v•w)u in R3. The Cross Product Def. v = ****3in R3. The cross product, v x w = **