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 GaussJordan 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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 Spring '08
 KLEMES
 Math, Linear Algebra, Vector Space, following line L1

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