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exam1SolutionsPctex

# exam1SolutionsPctex - Math 222 200 1(20 Define the...

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Math 222 - 200 Exam 1 Solutions February 25, 2005 1. (20) Define the following terms: (a) linearly independent set of vectors A set of vectors { ~x 1 , · · · , ~x k } is linearly independent if whenever λ 1 ~x 1 + · · · + λ k ~x k = ~ 0, then λ 1 = λ 2 = · · · = λ k = 0. (b) subspace A nonempty subset S of a vector space V over a field F is called a subspace if i. ~a and ~ b in S implies ~a + ~ b is also in S ii. α F and ~a S implies that α~a S . (c) dimension of a vector space The dimension of a vector space V is the number of vectors in a basis of V . (d) null space of a matrix If A is an m × n matrix, the null space of A is the set of vectors ~x R n (thought of as n × 1 matrices) such that A~x = ~ 0 R m . 2. (30) For the system of equations below: 2 x 1 - x 2 + 3 x 4 = 1 x 2 - x 4 = 6 Before answering any of the following questions the reduced row echelon form of the augmented matrix of this system is computed ˆ A = 2 - 1 0 3 1 0 1 0 - 1 6 = 1 0 0 1 7 / 2 0 1 0 - 1 6 (a) Find a particular solution. The given system is equivalent to the system x 1 + x 4 = 7 / 2 x 2 - x 4 = 6 . Setting x 3 = x 4 = 0, we have the particular solution h 7 / 2, 6, 0, 0 i . (b) Find a basis for the null space of the coefficient matrix. The coefficient matrix of the system is row equivalent to 1 0 0 1 0 1 0 - 1 . There are 2 free variables x 3 and x 4 , which tells us that the null space has dimension two. A basis for the null space is {h 0, 0, 1, 0 i , h- 1, 1, 0, 1 i} .

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(c) Find bases for the row and column spaces of the coefficient matrix.
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