midterm_sol_yrXXXX

midterm_sol_yrXXXX - 1.For what values of h the following...

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Unformatted text preview: 1.For what values of h the following system of linear equations has(1)unique,(2)in nitely many solutions,(3)no solution?(Each part values 10 points.) 8 x1 + x2 + x3 = 1 > > < 2 > 2x1 + x2 + hx3 = > : ?2x + hx ? hx = h ? 4 1 2 3 Answers: 2 3 32 11 1 1 111 1 42 1 h h?2 05 2 5 4 0 ?1 0 0 (h ? 2)(h + 1) h ? 2 ?2 h ?h h ? 4 (1) unique solution i h 6= ?1; 2; (2) in nitely many solutions i h = 2; (3) no solution i h = ?1. 2. Let T : R4 ! R4 be a linear transformation de ned by Is T invertible?If yes, nd its inverse linear transformation(20 pts). Midterm Solutions T (x1 ; x2 ; x3 ; x4 ) = (x1 + x2 ? x3 ? x4 ; ?x1 + x2 ? x3 + x4 ; x1 + x2 ? 2x3 ; x1 ? x2 ? 3x3 + 2x4 ): Answers: Yes. T ?1(x1 ; x2 ; x3 ; x4 ) = (?7=2x1 ? 3=2x2 +4x3 ? x4 ; ?5=2x1 ? 1=2x2 +3x3 ? x4 ; ?3x1 ? x2 +3x3 ? x4 ; ?4x1 ? x2 + 4x3 ? x4 ): by 3. Let T1 : R3 ! R3 ; T2 : R4 ! R4 ; T3 : R3 ! R4 ; T4 : R4 ! R3; be linear transformations de ned respectively.Which one of them is (a)one-to-one,but not onto;(b)onto,but not one-to-one;(c)both one-toone and onto,and (d)neither one-to-one nor onto?Please gives brief reason(40 pts). Answers: 2 ?1 A1 = 4 1 1 2 6 A3 = 6 4 2 3 12 1 62 3 0 5 ; A2 = 6 3 4 4 2 45 3 2 3 ?2 7 ; A4 = 4 1 1 7 11 05 12 0 32 4 1 57 60 76 65 40 7 0 32 11 1 2 25 40 12 0 3 7 7; 5 T1 (x1 ; x2 ; x3 ) = (?x1 + x2 + x3 ; x1 ? x2 + x3 ; x1 + x2 ? x3 ): T2 (x1 ; x2 ; x3 ; x4 ) = (x1 +2x2 +3x3 +4x4 ; 2x1 +3x2 +4x3 +5x4 ; 3x1 +4x2 +5x3 + x6 ; 4x1 +5x2 +63x3 +7x4 ): T3 (x1 ; x2 ; x3 ) = (x1 + 2x2 + 3x3 ; 2x1 + 3x2 + 4x3 ; 3x1 + 4x2 + 5x3 ; 3x1 + 2x2 + x3 ): T4 (x1 ; x2 ; x3 ; x4 ) = (x1 + x2 + x3 + x4 ; x1 + x2 + 2x3 + 2x4 ; x1 + 2x2 + x3 + 2x4 ): 32 11 ?1 1 ?1 1 5 4 0 2 1 ?1 00 32 123 12 2 3 4 7 6 0 ?1 76 3 4 55 40 0 321 00 1. No one is (a)one-to-to,but not onto; 2. A4 is (b)onto,but not one-to-one; 3 4 5 6 2 ?1 0 0 4 ?3 0 0 3 111 1 0 1 5: 011 3 ?2 0 0 3. A1 is (c)both one-to-one and onto; 4. A2 and A3 are neither one-to-one nor onto. 4. Circle only YES or NO;no reason needed(20 pts). Let A be an mxn matrix and T : Rn ! Rm a linear transformation de ned by T (x) = Ax: (a) If the column vectors of A are independent, then T is one-to-one. (b) If the row vectors of A are independent, then T is one-to-one. (c) If the column vectors of A are independent, then T is onto. (d) If the row vectors of A are independent, then T is onto. (e) If every row of A has a pivot position, then T is one-to-one. (f) If every column of A has a pivot position,, then T is one-to-one. (g) If every row of A has a pivot position, then T is onto. (h) If every column of A has a pivot position,, then T is onto. (i) If v1 ; ; vk are linearly independent vectors in Rn, then T (v1 ); ; T (vk ) are linearly independent. (j) If v1 ; ; vk are linearly dependent vectors in Rn , then T (v1 ); ; T (vk ) are linearly dependent. Answers: (a) Yes (b) No (c) No (d)Yes (e)No (f)Yes (g)Yes (h)No (i)No (j)Yes. ...
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This note was uploaded on 01/28/2011 for the course MATH 113 taught by Professor Beifangchan during the Fall '08 term at HKUST.

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