Midterm_Sol - Math 104 Midterm Instructions Complete the...

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Math 104 : Midterm Instructions: Complete the following 4 problems. Remember to show all your work. No notes or calculators are allowed. Please sign below to indicate you accept the honor code. Name: SUID: Signature: Problem 1 2 3 4 Total Score
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Problem #1. (20 pts) Let w 1 , w 2 and w 3 be three vectors in C 3 . Let v 1 = w 1 - w 3 , v 2 = w 1 + w 2 , v 3 = w 1 + λ w 3 , and v 4 = 2 w 1 + w 2 - w 3 . Where here λ C . For what value λ 0 is it always true that when λ = λ 0 , v 1 , v 2 , v 3 and v 4 never span C 3 . Justify your answer. (Hint: Rewrite the problem using matrices). Answer: Let us set V = v 1 | v 2 | v 3 | v 4 and W = w 1 | w 2 | w 3 Then we have V = WA where A = 1 1 1 2 0 1 0 1 - 1 0 λ - 1 The v i do not span C 3 when and only when dimR ( V ) 2. By the rank- nullity theorem this occurs when and only when dimN ( V ) 3. Notice that N ( A ) N ( V ) and so dimN ( A ) dimN ( V ). Applying the Gaussian elim- ination algorithm to A one arrives after a sequence of row operations to A with A = 1 0 1 1 0 1 0 1 0 0 λ + 1 0 Notice if λ + 1 = 0 then rref ( A ) has 2 pivots. Otherwise rref ( A ) has 3 pivots. In the former case, dimN ( A ) = dimN ( rref ( A )) = 4 - 2 = 2 while in the latter dimN ( A ) = dimN ( rref ( A )) = 4 - 3 = 1. In particular, λ 0 = - 1 always ensures that the v i do not span. Notice that if the w i are linearly independent and λ 0 = - 1 then dimN ( V
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