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quiz1_1806_s05

# quiz1_1806_s05 - A into LU =(lower triangular(upp er...

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18.06 Professor Strang Quiz 1 February 28, 2005 Grading 1 Your PRINTED name is: 2 3 4

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1( 2 6 p t s . ) Suppose A is reduced by the usual row operations to 1402 R = 0012 . 0000 Find the complete solution (if a solution exists) to this system involving the original
2( 1 8 p t s . ) Suppose the 4 by 4 matrices A and B have the same column space .Th e y may not have the same columns ! (a) Are they sure to have the same number of pivots ? YES NO WHY? (b) Are they sure to have the same nullspace ? YES NO WHY?

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3( 4 0 p t s . ) (a) Reduce A to an upper triangular matrix U and carry out the same elimination steps on the right side b : 33 1 b 1 ± ² ± ² Ab = 35 1 b 2
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Unformatted text preview: A into LU = (lower triangular)(upp er triangular). A (b) If you change the last entry in A from 2 to (what number gives A new ?) then new becomes singular. Describe its column space exactly . A (c) In that singular case from part (b), what condition(s) on b 1 , b 2 , b 3 allow the system new x = b to be solved ? ⎡ ⎤ 3 ⎢ ⎥ ⎢ ⎥ (d) Write down the complete solution to A new x = ⎢ 3 ⎥ (the ﬁrst column). ⎣ ⎦ − 3 4 4 (16 pts.) Suppose the columns of a 7 by 4 matrix A are linearly independent. (a) After row operations reduce A to U or R , how many rows will be all zero (or is it impossible to tell) ? (b) What is the row space of A ? Explain why this equation will surely be solvable: ⎡ ⎤ 1 ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ A T y = . ⎢ ⎥ ⎢ 0 ⎥ ⎣ ⎦ 0 5...
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quiz1_1806_s05 - A into LU =(lower triangular(upp er...

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