Find a basis for the set of vectors in bbb(R)3 in the plane. x-6y+9z=0.



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Find a basis for the set of vectors in R3 in the plane x6y+9z=0.[Hint: Think of the equation as a​ "system" of homogeneous​ equations.]
A basis for the set of vectors in R3 in the plane x6y+9z=0

Answer & Explanation



Skilled2021-09-02Added 91 answers

Consider the plane x6y+9z=0 as system of homogeneous equation Ax=0.
[1 6 9][xyz]=0

Thus, the null space of the above matrix [1 6 9] is the basis for the set of vectors R3 in the given plane x6y+9z=0. Solve the equation x6y+9z=0 for x as follows.


Therefore, the vector x in terms of free variables can be written as,



That is, the set of vectors [610],[901]a basis for null space of A. 

Thus, the vasis for the set of vectors R3 in the given plane

x6y+9z=0 is [610],[901].

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