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Linear Algebra Basis for Hyperplane - Example 8 Explanation
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[QUOTE="brmath, post: 4525875, member: 486151"] The first step is to determine whether the given vectors, which I'll call ##v_1, v_2, v_3## are independent. You can stack them up into a 4 x3 matrix and triangularize it. If you wind up with a zero row, they are not independent. But suppose they are independent. For them to be a basis you must be able to write any point ##(x_1,x_2,x_3,x_4)## in the hyperplane as a linear combination fo the v's. That is there must be numbers a, b, c such that ##(x_1,x_2,x_3,x_4)## = ##av_1 + bv_2 + cv_3##. Now how do you know that ##(x_1,x_2,x_3,x_4)## is a vector in the hyperplane? Well, you solve the hyperplane equation for ##x_4 \text { in terms of } (x_1,x_2,x_3)## then plug in the ##x_4## you got into the 4th position. By doing this you incorporated the requirements of the hyperplane into your vector. Can you start now? If not, ask some specific questions. [/QUOTE]
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Linear Algebra Basis for Hyperplane - Example 8 Explanation
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