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Determine whether the following sets are subspaces of R3 under the operations of

Chapter 1, Problem 8

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QUESTION:

Determine whether the following sets are subspaces of R3 under the operations of addition and scalar multiplication defined on R3 . Justify your answers. (a) W) = {(01,02,03) R3 : ai = 3a2 and 03 = a2) (b) W2 - {(01,02,03) R3 : a, = a 3 + 2} (c) W3 = {(01,02,03) R3 : 2ai - 7o2 + a 3 = 0} (d) W4 = {(ai,a 2 , a3) R3 : O] - 4a2 - a 3 = 0} (e) W5 = {(aj, a2,03) R3 : ai + 2a2 - 3o3 = 1} (f) W6 = {(01, a 2 , a3) R3 : 5a? - 3a| + 60^ = 0}

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QUESTION:

Determine whether the following sets are subspaces of R3 under the operations of addition and scalar multiplication defined on R3 . Justify your answers. (a) W) = {(01,02,03) R3 : ai = 3a2 and 03 = a2) (b) W2 - {(01,02,03) R3 : a, = a 3 + 2} (c) W3 = {(01,02,03) R3 : 2ai - 7o2 + a 3 = 0} (d) W4 = {(ai,a 2 , a3) R3 : O] - 4a2 - a 3 = 0} (e) W5 = {(aj, a2,03) R3 : ai + 2a2 - 3o3 = 1} (f) W6 = {(01, a 2 , a3) R3 : 5a? - 3a| + 60^ = 0}

ANSWER:

Step 1 of 6

(a)

The subset is given as,

                                           

It is known that the W is a subspace of vector space V if W has a zero vector,  whenever  and  and whenever  and .

The zero vector belongs to  as,

                                              

The sum of two vector and scalar product is given as,

 ,  and

Hence, the set  is a subspace of .

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