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Unformatted text preview: Math 136 Additional Practice for Term Test 1 1 5 1 1 , −1 , 1 . 1: Let B = −2 3 0 0 (a) Determine if v = 5 is in span B . 5 (b) Determine if B is linearly independent or linearly dependent. −1 1 2: Let v = 1 and w = −1 2 3 (b) Calculate projv w . (c) Calculate v × w . (a) Determine if v and w are orthogonal. (d) Find a scalar equation of the plane with vector equation x = sv + tw , s, t ∈ R. 3: Determine, with proof, which of the following sets is a subspace of R2 . (a) S1 = (b) S2 = x1  (x1 + x2 )2 = 0 . x2 x1  2x1 + 3x2 = 0 . x2 4: Suppose that the set {v1 , v2 , v3 } is linearly independent in Rn . Determine, with proof, if {v1 , v1 + v2 , v1 + v2 , v3 } is linearly independent or linearly dependent. 1 2 L2 5: Consider three planes P1 , P2 , and P3 , with respective equations a11 x1 + a12 x2 + a13 x3 = b1 , a21 x1 + a22 x2 + a23 x3 = b2 , and a31 x1 + a32 x2 + a33 x3 = b3 . The intersections of these planes are illustrated in the diagram. Assume that L1 , L2 , and L3 are parallel. a11 a12 (a) What is the rank of the matrix A = a21 a22 a31 a32 a11 a21 (b) What is the rank of the matrix [A  b] = a31 a13 a23 ? Explain. a33 a12 a13 b1 a22 a23 b2 ? a32 a33 b3 L1 P1 P3 P2 L3 ...
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 Spring '08
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 Linear Algebra, Algebra, Addition

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