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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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 Spring '05
 Strang
 40 pts, 16 pts, 18 pts, 26 pts, Professor Strang

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