MATH 4130 Midterm: Exam 1 Solution
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Math 413/513 - fall 2010: problem 1. Solution. dim(v ) = 3 since v = span{v1, v2, v3} and the vectors v1, v2, v3 are lin. independent. Same argument for w : dim(w ) = 3 since w1, w2, w3 are lin. ind. (a similar argument is needed!) (b) find a basis for the sum v + w . Since {v1, v2, v3} is lin. ind. and w1 / span{v1, v2, v3} (this needs proof!) then, by theorem. 1. 7, {v1, v2, v3, w1} is lin. ind. in r4. Now {v1, v2, v3, w1} v + w means that v + w = r4. We conclude that {v1, v2, v3, w1} is a basis for v + w . (c) find a basis for the intersection v w . Verify that dim(v + w ) + dim(v w ) = dim(v ) + dim(w ). Hence a"s and b"s are solutions to the underdetermined linear system a1.
