# Let W be the subspace of R3 spanned by { [1, 2, 4], [-1, 2, 0], [3, 1, 7]}.(a) Find a basis for W perpendicular.b) Find dimW and dimW perpendicular.(c) Describe W and W perpendicular geometrically.

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by sweetysiri92

2014-11-04T01:28:37+05:30

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Form a matrix A from the 3 given vectors in the space of W.

W perpendicular is orthogonal to W, means  every vector in that space is orthogonal to every vector in W.

This is possible when A X = O

Find row reduced Echelon form of A :

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Dimension of W = rank of A = number of independent rows in A = 2

Null space is the orthogonal space (orthogonal complement) of A,  It has [ 1, 2, 4 ].
Its dimension = 1   ( = n - dimension of W)

The operations performed on 1st row to become O are : 7 row1 - 4 Row2 + 2 Row3
= 7 * [2, 0, 4 ] - 4 * [ -1, 2, 0 ] + 2 [ 3, 1, 7 ]

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W is a plane in in three dimensional space.  x = 2 y = - 2 z
(it passes through origin)

W perpendicular is the Line perpendicular to the plane of W.

thanx a lot. u r welcom