wksht10 - MATH 152 L1 & L2 Applied Linear Algebra and...

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Unformatted text preview: MATH 152 L1 & L2 Applied Linear Algebra and Differential Equations Week 10 Worksheet (April 11, 2005): Euclidean Vector Spring 2004-05 Q1. (Linear combination of vectors) Consider the vectors v1 = (1, 3, 0), v2 = (−1, 1, 1), v3 = (3, 1, −1). Is b = (3, 5, −1) a linear combination of v1 , v2 , v3 ? Q2. (General solution in vector form) Consider 20 2 1 A = 1 1 − 1 0 , 0 1 −2 1 2 b = −1 . −2 (a) Find a special solution x = (x1 , x2 , x3 , x4 ) of Ax = b satisfying x1 = x2 = 0. (b) Solve the homogeneous equation Ax = 0. (c) Solve the equation Ax = b. Q3. (Free variables and rank) Suppose Ax = b is a consistent system of 5 equations in 9 variables. If the rank of A is 3, how many free variables are there in the general solution? Q4. (Null space) Find the null space of A if −3 A= 1 2 6 −2 −4 −1 2 5 1 3 8 −7 −1. −4 Q5. (Span) Consider the vectors v1 = (1, 2, 3, 4), v2 = (2, 3, 4, 1), v3 = (3, 4, 1, 2). (a) Is b = (0, 0, 1, −1) in the span of v1 , v2 , v3 ? (b) Do v1 , v2 , v3 span R4 ? If not, how many more vectors do you need to add in order to span R4 ? Explain. Q6. (Span) Consider the vectors v1 = (0, 1, 2, −1), v2 = (1, 4, −3, 1), v3 = (2, 3, 1, 1). v4 = (−1, 3, 0, 2). Determine whether the vector b = (1, 0, 6, −1) is in the span of v1 , v2 , v3 , v4 . If b is indeed in the span of v1 , v2 , v3 , v4 , how many vectors (among v1 , v2 , v3 , v4 ) can we remove such that b is still in the span of the remaining vectors? Q7. (Column space) Determine whether the vector b is in the column space of A. 2 4 −2 1 3 7 3 , A = −2 −5 b = − 1 . 3 7 −8 6 3 Q8. (Linear independence) Suppose the vectors v1 , v2 , · · · , vn (n Determine if the following vectors are also linearly independent. (a) The vectors v1 − v2 , v2 − v3 , v3 − v1 . 4) are linearly independent. (b) The vectors v1 − v2 , 2(v2 − v3 ), 3(v3 − v4 ), · · · , n(vn − v1 ). (c) The vectors v1 , v1 + c1 v2 , v2 + c2 v3 , v3 + c3 v4 , · · · , vn + cn v1 , where c1 , c2 , · · · , cn ∈ R. ...
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This note was uploaded on 09/30/2010 for the course MATH MATH152 taught by Professor Kcc during the Spring '10 term at HKUST.

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wksht10 - MATH 152 L1 & L2 Applied Linear Algebra and...

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