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Unformatted text preview: Assignment 6 Math 417 â€” Winter 2009 Due March 6 Â§ 3.2#46. The kernel of this matrix A consists of all the solutions to the homogeneous linear sytem A~x = ~ 0. To find a basis for the kernel, begin by solving that linear system by rowreduction. That needs no work, since A is already in rref (by the kindness of the bookâ€™s author). So we can read off the solution: x 2 , x 4 , x 5 arbitrary, x 1 = 2 x 2 3 x 4 5 x 5 , and x 3 = 4 x 4 6 x 5 . In other words, the general solution of the system (i.e., the general vector in ker( A )) is  2 x 2 3 x 4 5 x 5 x 2 4 x 4 6 x 5 x 4 x 5 = x 2  2 1 + x 4  3 4 1 + x 5  5 6 1 . Thus, the three vectors displayed in this last equation span ker( A ). Furthermore, they are linearly independent (by inspection of their second, fourth, and fifth components). So they form a basis for ker( A ). Â§ 3.2#48. The given equation defining the plane V can be written as the matrix equation 3 4 5 x 1 x 2 x 3 = [0] . This means that V is the kernel of 3 4 5 . To express V as the image of a matrix, we first solve the defining equation (for example, by reduction to rref, which takes just one step). The general solution, i.e., the general point in V , is x 1 x 2 x 3 =  4 3 x 2 5 3 x 3 x 2 x 3 = x 2  4 3 1 + x 3  5 3 1 =  4 3 5 3 1 1 x 2 x 3 . This means that V is the image of the matrix  4 3 5 3 1 1 . 1 Â§ 3.2#50. Yes, V + W is a subspace. Iâ€™ll check the bookâ€™s definition of subspace. First of all, since V and W are subspaces, we have ~ âˆˆ V and ~ âˆˆ W and therefore ~ 0 = ~ 0 + ~ âˆˆ V + W . Second, Iâ€™ll show that V + W is closed under multiplication by numbers. Consider any vector in V + W , say ~v + ~w where ~v âˆˆ V and ~w âˆˆ W , and consider any number r . Since V and W are subspaces, we know r~v âˆˆ V and r~w âˆˆ W , and therefore r Â· ( ~v + ~w ) = r~v + r~w âˆˆ V + W. Finally, Iâ€™ll show that the sum of any two vectors from V + W is again in V + W . Let the two vectors be ~v + ~w and ~v + ~w , where ~v,~v âˆˆ V and ~w, ~w âˆˆ W . Since V and W are subspaces, we know ~v + ~v âˆˆ V and ~w + ~w âˆˆ W . Therefore, ( ~v + ~w ) + (...
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 Spring '09
 eveet
 Math, Linear Algebra, Algebra, Vectors, Vector Space

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