a) Determine a condition under which (x, y, z) is a linear combination of [-3, 5, -3], [-9, 11, -3],

klancimarsn

klancimarsn

Answered question

2022-02-23

a) Determine a condition under which (x, y, z) is a linear combination of [-3, 5, -3], [-9, 11, -3], [-6, 8, -3]? Your condition should take the shape of a linear equation. I have the theorem that every vector (x, y, z) in R3 is a linear combination of x(1,0,0)+y(0,1,0)+z(0,0,1).

Answer & Explanation

Pregazzix2a

Pregazzix2a

Beginner2022-02-24Added 9 answers

In general we need to write down the system 
av1+bv2+cv3=(x,y,z) 
in augmented matrix form and handle it by RREF to find the condition on x,y,z for the system to have solution. 

This approach provides the cartesian equations of the subspace spanned by v1,v2,v3
As an alternative look for the dimensions of the subspace, if you find dim=3 then the linear combination always has solution. Otherwise select a basis and solve the system for the basis vectors.

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