Math 300
Laboratory 3 โ Vector Spaces
Due Tuesday, October 27, 2009 โ 10 points
Names:
You may use books and notes in doing this lab but please work with your partner only. Each group turns
in one solution.
1. (5 points) Let Vbe the set of all pairs .x; y/ of real numbers such that y>0. For example,
.2; 3/; .๎1; 5/ 2Vbut .4; ๎1/; .0; 0/ โฆV. Define addition and scalar multiplication for Vas
follows:
.x1;y
1/C.x2;y
2/D.x1Cx2;y
1y2/and c.x; y/ D.cx; y c/: (1)
For example, .2; 3/ C.๎5; 4/ D.2 C.๎5/; 3 ๎4/ D.๎3; 12/,2.5; 3/ D.2 ๎3; 32/D.6; 9/, and
๎2.0; 4/ D.๎2๎0; 4๎2/D.0; 1
16 /. Prove that Vwith the operations defined in (1) is a vector
space.
When checking the axioms, you may use the familiar properties of real numbers from high school
algebra, for example, xy Dyx and .xy/cDxcyc.
2. (5 points) Let M2;2 be the vector space of 2๎2matrices. Consider the following set Sof vectors
in M2;2.
SD(๎13
2
21
๎;๎๎12
11
3๎;"122
3
7
6๎265
9#;"2
31
1
2๎4#):
(a) Is Slinearly independent? Justify your answer. If Sis linearly dependent, express some
vector in Sas a linear combination of the other vectors in S.
(b) Is Sa spanning set for M2;2? Justify your answer.
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