Question

(a) (5 points.) Let W CW CW CW3 be distinct subspaces of R? True/False (Justify your answers): (i) Wo must be the zero subspace. (ii) W, must be R. (iii) W, must be RP. (iv) Suppose V1, V2, V3 are vectors such that vi EWW -for each 1 <i<3. Then {V1, V2, V3} must be a basis for R. (v) There are three linearly independent vectors in R that do not form a basis for R?

Answer 1

0 Answer: Let woch, ciecos be detinct subspaces. of Rs (2) Dimension of $?983. so posible Misspae of PR have dimension 91,2,3. Also, then t he conteinmente Henes, wo must-se o dimeritonal wo es zu tusspall. seaux soppose This happens ACD au tro Musspan and dem B= 2, die A-3 such thing not ponose. 19 din As dem B. Her, the contarment de strictly : die Acum B. Olong the den worden wederaten den log (lo, SIR") I OCI243. This given statement is true be 2 Verorsional and (1) By asore logie con must os og C TR ? 80th have domesoon 3. W =R? . It is tere statement

( cauld be constfen as BEW, & Wz de not exactly R. It has dimensions 10, 310 morphe cooth R. It is time statement. D, Ew but we wo=20} vs twe; Us CW, = could be linear combination of V. So, &, 43 linearly brindependent. Sinclarly, de EWz, de fles & spanade, den? {1, V2, 23 } lineady independent. 1468, they are non number and they grans dimensional susspace of R. Hence this must span R as dem R=3. tv, 12, 18, 8014 of It is ten statement. The linearly independent must span R? DR has Vernoon 3. Theutone Mit fare statement.