Question

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.

Answer 1

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

$A = \left[\begin{array}{ccc}1&2&4\\-1&2&0\\3&1&7\end{array}\right],\ \ \ Let\ X= \left[\begin{array}{c}x_1&x_2&x_3\end{array}\right]$ 

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 :

$A= \left[\begin{array}{ccc}1&2&4\\-1&2&0\\3&1&7\end{array}\right] = \left[\begin{array}{ccc}2&0&4\\-1&2&0\\3&1&7\end{array}\right]=\left[\begin{array}{ccc}2&0&4\\-1&2&0\\0&1&1\end{array}\right]\\\\=\left[\begin{array}{ccc}2&-4&0\\-1&2&0\\0&1&1\end{array}\right]=\left[\begin{array}{ccc}0&0&0\\-1&2&0\\0&1&1\end{array}\right] \\\\A X=O\\\\x_2+x_3=0\\-x_1+2x_2=0\ \ or,\ x_1=2x_2=-2x_3\\\\X=[1, 2, -2]^T$
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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.