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mockmathMT

# mockmathMT - A:= 1 2 3-1 1-2 0 1 1 and B:= 2 1 0-1 1 1 3 2...

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MATH 54, mock midterm test. Name Student ID # All the necessary work to justify an answer and all the necessary steps of a proof must be shown clearly to obtain full credit. Partial credit may be given but only for significant progress towards a solution. Show all relevant work in logical sequence and indicate all answers clearly. Cross out all work you do not wish considered. 1 page of notes is allowed. Books and electronic devices are not allowed during the test. 1. Is the linear system x 1 + x 2 = 0 x 1 - 3 x 2 = 0 x 1 + 2 x 2 = 1 solvable? What is its least-squares solution? Is it unique? 2. Given the matrix A = 1 2 3 - 1 1 - 2 0 1 1 . (a) compute its determinant. (b) Is the matrix A T A invertible? 3. Are the matrices

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Unformatted text preview: A := 1 2 3-1 1-2 0 1 1 and B := 2 1 0-1 1 1 3 2 1 (a) row equivalent? (b) similar? 4. Show that he formula h f, g i := 1 2 π Z 2 π f ( t ) g ( t ) dt deﬁnes an inner product on C [0 , 2 π ]. Let W := span { 1 , sin t, cos t } . What is the orthogonal projection of t onto W ? 1 5. A linear map T : IP 2 → IP 3 is deﬁned as T : f ( t ) 7→ ( t 2 f ( t )) . Find (a) its matrix representation for the standard bases { 1 , t, t 2 } of IP 2 and { 1 , t, t 2 , t 3 } of IP 3 ; (b) a basis for ker( T ); (c) a basis for range( T ). 6. Given A := ± 1 2 1 0 ² , compute A 1000 . 2...
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mockmathMT - A:= 1 2 3-1 1-2 0 1 1 and B:= 2 1 0-1 1 1 3 2...

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