two_i_a_allproofs.pdf - Spaces and their subspaces Linear...

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Spaces and their subspaces Linear Algebra Jim Hefferon
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Real three-space
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Subspaces of R 3 The next slide gives a sample of subspaces of the vector space R 3 : the entire space, planes, lines, and the trivial subspace. Subsets are drawn connected to their supersets on the levels directly above and below. On the level one up from the bottom, the second and third subspaces are lines because the conjunction of the two conditions means that each is the intersection of two planes.
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R 3 xy -plane { x y z | x + y + z = 0 } { x y z | x + 2z = 0 } . . . y -axis { x y z | x - y + z = 0 and x + 2z = 0 } { x y z | y = 2x and z = 0 } . . . trivial subspace { 0 0 0 }
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Express subspaces as spans Example Consider the plane. P = { x y z | x + y + z = 0 } Take the condition x + y + z = 0 as a one-equation linear system and parametrize. P = { - y - z y z | y, z R } = { - 1 1 0 y + - 1 0 1 z | y, z R } Thus the plane is this span. P = [ { - 1 1 0 , - 1 0 1 } ]
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This shows P . The two vectors from the spanning set are in red. For each, its body lies in the plane.
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Example For the plane ˆ P = { x y z | x + 2z = 0 } repeat the process ˆ P = { - 2z y z | y, z R } = { 0 1 0 y + - 2 0 1 z | y, z R } to express it as a span. ˆ P = [ { 0 1 0 , - 2 0 1 } ]
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Example For the plane ˆ P = { x y z | x + 2z = 0 } repeat the process ˆ P = { - 2z y z | y, z R } = { 0 1 0 y + - 2 0 1 z | y, z R } to express it as a span.
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