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Unformatted text preview: TEST 1 (Math 250 A) l. (a) Show that the midpoint of P1 = (x1,y1) and P2 = (x2 , yz) is given by: (20 pts) PM =£xlgx2 ’y1 'lz'yzj (Hint: Consider any of the triangles generated by the position vectors 0131 , 0P2 and 0PM ) (lo) Use the above formula to ﬁnd the midpoint of P1 = (1,—2) and P2 = (—3 ,4) . 2. (21) Find the equation of the line L through the points P1 = (1,0,—l) and P2 = (2,3,—1) .
Does the point Q = (0,1,—1) lie on L ? (20 pts) (b) Find the equation of the plane M that contains the point P = (1,—1, 2) and the line
L: x=t,y=l+t,z:—3+2t. 3. The set 1R+ = {x e R  x 2 0} , equipped with the following addition and scalar multiplication: (20 pts)
x + y = xy xi ' x = x’1
becomes a vector space. In this vector space, show that: (a)l+2=2 ' (b)0.2=1 4. Let W = {p(x) E Pix]  19(3) = 0}. Show that Wis a subspace of P2[x] . (10 pts)
(Note: Wis the set of all quadratic polynomials that have 3 as a root) 5. Let W = {A 6 MM (R)  det(A) = O} . ls Wa subspace of M2X2 ( i) ? (10 pts) If your answer is no, provide a counterexample. (Note: Wis the set of all noninvertible matrices) 6. Show that the vectors x2 + x3 ,x, 2x2 + 1,3 span the vector space P3[x] . (10 pts) 1 l l 0
7. Show that the vectors A1 = [0 ],A2 =[ j are linearly independent in M M (R). (10 pts) ...
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 Spring '07
 Aristidou

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