Math133McGillpracticeexam

Math133McGillpracticeexam - MATH 133 - Practice Exam x = 3t...

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Unformatted text preview: MATH 133 - Practice Exam x = 3t + 2 y = −2t + 1 , and the point A(1, −1, 2), Find the 1. Given the following line L1 : z=t equation of the plane perpendicular to L1 and passing by A. x = 3t + 2 2. Find the distance between the line L1 : y = −2t + 1 , and the point A(1, −1, 2). z=t 3. Find the equation of line AB given that A(1, 2, 3) and B (−2, 0, 2). 4. Find a vector perpendicular to the plane ABC given that A(1, 2, 3), B (−2, 0, 2), and C (2, 4, −1). x = 3t + 2 y = −2t + 1 find the intersection of P1 5. Given (P1 ) : 4x − 3y + z = 2 and L1 : z=t and L1. 6. Find the solution of the following system using Gauss-Jordan elimination 4x1 + x2 − 3x3 + x4 = 1 −x1 + 3x2 − x3 + 2x4 = 7 k x − 2y = 3 have 2x − ky = 4 7. For what value(s) of k does the system a) No solution. b) An infinite number of solutions. c) A unique solution. 8. Given A = 2 4 −1 −3 a) Write A−1 as a product of elementary matrices. b) Write A as a product of elementary matrices. 9. Find the dimension of the following subspaces of Rn 1 2 4 −1 a) span 0 , 1 , 1 , −1 −1 1 −1 −2 1 1 −1 1 , 0 , 1 b) span 1 2 0 10. Given that 1 abc def ghi = −2 Find −a d 2g a) −b e 2h −c f 2i b) 3a 3b 3c g + 3a h + 3b i + 3c 2d 2e 2f c) abc 2d 2e 2f def 11. Find the determinant of the following matrix 1 0 2 −3 0 2 1 −2 1 0 0 1 1 0 A = −1 0 −2 1 −1 1 0 0 1 −1 0 1 12. Find an orthogonal basis for R3 containing the vector −1 2 13. Find W⊥ given each of the following cases 1 2 a) W = span −1 , 0 2 1 1 0 b) W = span 2 −1 1 0 −1 14. Given A = 0 1 0 −1 0 1 a) Find an orthogonal matrix P and a diagonal matrix D such that D = P T AP . 15. Find the matrix of the linear transformation for a) the counterclockwise rotation about the origin by and angle of π . 3 b) the relfection about the line 3x1 − 2x2 = 0. c) the projection on the line 4x1 − x2 = 0. 2 16. Are the vectors w,u, and v independant? 1 5 3 a) w = −2 , u = 6 , v = 2 . 3 −1 1 1 2 0 0 , u = −1 , v = 2 . b) w = 3 1 1 3 ...
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