Question

Determine whether the following expressions are subspaces or affine subspaces and give the reasoning: a) union if plane xy and the z axis, b) x-2y+z=2, c) the line given by the equation system x=4+2t, y=2+t, z=-6-3t

          Determine whether the following expressions are subspaces or affine subspaces and give the reasoning: a) union if plane xy and the z axis, b) x-2y+z=2, c) the line given by the equation system x=4+2t, y=2+t, z=-6-3t
        

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Elementary and Intermediate Algebra
Elementary and Intermediate Algebra
Alan S. Tussy, R. David Gustafson 5th Edition
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Determine whether the following expressions are subspaces or affine subspaces and give the reasoning: a) union if plane xy and the z axis, b) x-2y+z=2, c) the line given by the equation system x=4+2t, y=2+t, z=-6-3t
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Transcript

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00:01 Okay, so let's get started with a.
00:03 Well, in this case, w is not a subspace.
00:11 How can we see this? well, the vector 1 .1 clearly belongs to w because the coordinates of this vector here are greater than or equal to 0.
00:26 But the vector negative 1 multiplied by 11 does not belong.
00:33 To w because this guy here is equal to the vector negative one negative one perfect so we showed that w is not a subspace now for part b of this exercise well in this case t is a subspace how can we see this well we just need to notice that t is defined by the following homogeneous equation t is defined by the equation z minus x minus 2 y equals 0 perfect and well since t is the set of solutions of a homogeneous linear equation this guy is a a subspace...
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