Question

Preview Activity 4.1.1. Before we introduce the definition of eigenvectors and eigenvalues, it will be helpful to remember some ideas we have seen previously. a. Suppose that v is the vector shown in the figure. Sketch the vector 2v and the vector - v. V b. State the geometric effect that scalar multiplication has on the vector v. Then sketch all the vectors of the form lv where is a scalar. C. State the geometric effect of the matrix transformation defined by [: _:] d. Suppose that A is a 2 x 2 matrix and that vi and v2 are vectors such that Av1 = 3v1, Av2 = -V2. Use the linearity of matrix multiplication to express the following vectors in terms of vị and v2. i. A4v1) ii. A(V1 + V2) iii. A(4v1 – 3v2). iv. APv1 V. A(4v1 - 3v2). vi. A"v1

Answer 1

I have done it for you in detail. Kindly go through.

a) zu: 5 2V 3 2 2 3 tra -2 a b) If 2 is positive scalar, av is the rector of magnitude 121 times the magnitude of vector direction will be reversed when aco and the same for ? so and yielde o when A=o. Geometric interpretation is that it stretches, or contracts, vectors will remain constant factor. by

c) 3 6 an [o ] O O 3 0 to 0 -1 [:-] and 1 maps maps 11 1:0) - ::] [::] [ ] is a stretching in x-axis by a factor of 3, is reflections along a combination of stretching factor of 3 in x-axis and reflecting We See and [60] x-axis. 3 O is by a factor along x. x-axis 2 where A= (30) Ava o -2.o 2 3 Av₂ 1-2 We Se what happens . to unit square in R² court (OP) (3,07 (0,-1) (3,-1)

Av, = 3v, an and A₂ - ₂ () A (V+V2) Cri) A( 4V, -342) = 3 (1) A (4 v) 4 A (",) 4 (3v,) = 12V, A(M) + A(V2) 3v, - V2 A (4 V, )+ A(-342) = 4 A() - 3 A(V2) - 4 (3 v) -3 (-42) 12 v, +372 A(Av.) = A ( 3v.) = 3 A(v.) 3 (34,)=9Y (u) A² ( 40,-342) = A (A (4V, -3v.)) - A(12V, +3va) from (ili) = A (12 V,) + Alva) 12 A(V.) + 3 A(N2) 12 (3v,) +36-V2) (iv) A?v, 36 v, -3₂ (vi) Aiv, A® (AV) - A°(3.) APA(3v.) A 3 ACV) A”(3.3v,) - A* (av) - A (A(902) - Aſa A (w)) = A (9.3v) A(27v.) - 27 ACV) = 27.3V 81 Vo