# tst1sltn fall 2015.pdf - MAT343 FALL 2015 TEST1 SECTION 1...

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MAT343, FALL 2015, TEST1, SECTION 1 SOLUTIONS Name: (BLOCK LETTERS PLEASE!) Do NOT write your answers on this sheet please! Grades: A=35+, B=25+, C=20+ D=15+ Problem 1: Compute the intersection of the three planes in R 3 , given by the equations x - 2 y + z = 0 , x + z = 1 , x + 2 y + z = - 2 . Describe the their relative location to each other (i.e. do they intersect, are they parallel?). SOLUTION: 1 - 2 1 0 1 0 1 1 1 2 1 - 2 , R 2 = R 2 - R 1 , R 3 = R 3 - R 1 , 1 - 2 1 0 0 2 0 1 0 4 0 - 2 , R 3 = R 3 - 2 R 2 1 - 2 1 0 0 2 0 1 0 0 0 - 4 There is no intersection. None of the three planes are parallel. Problem 2: Find bases for the nullspace and the column space of the matrix A = ( - 5 1 - 4 2 - 1 1 - 2 0 ) . SOLUTION: ( - 5 1 - 4 2 - 1 1 - 2 0 ) x = 0 , R 2 = 5 R 2 - R 1 , ( - 5 1 - 4 2 0 4 - 6 - 2 ) x 3 = s, x 4 = t, x 2 = 3 2 s + 1 2 t, x 1 = - 1 2 s + 1 2 t, x = sb 1 + tb 2 , b 1 = - 1 2 3 2 1 0 , b 2 = 1 2 1 2 0 1 Problem 3: Compute the determinant of the matrix 1 x x 2 1 y y 2 1 z z 2 . 1 This study resource was shared via CourseHero.com Subscribe to view the full document.

For what values of x, y, z is this matrix singular? SOLUTION: Unformatted text preview: 1 x x 2 y-x y 2-x 2 z-x z 2-x 2 , det = ( y-x )( z 2-x 2 )-( z-x )( y 2-x 2 ) = ( y-x )( z-x )( z + x-y-x ) = ( y-x )( z-x )( z-y ) Singular for x = y or y = z or z = y Problem 4: Find the distance between the plane given by x-2 y + z = 0 and the point with coordinates p = (1 , , 0) SOLUTION: Basis for the subspace given by the plane: v = sb 1 + tb 2 , b 1 = 2 1 , b 2 = -1 1 , ( b T 1 b T 2 ) ( p-q ) = 0 , q = ( b 1 , b 2 ) ( s t ) B T B ( s t ) = B T p, s = 1 3 , t =-1 6 , q = . 8333 . 3333-. 1667 , p-q = . 1667-. 3333 . 1667 , d = | p-q | = 0 . 4082 2 This study resource was shared via CourseHero.com Powered by TCPDF ( )...
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