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&nbsp;R3&nbsp;is a linear combination of&nbsp;x(1,0,0)+y(0,1,0)+z(0,0,1).

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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&amp;nbsp;R3&amp;nbsp;is a linear combination of&amp;nbsp;x(1,0,0)+y(0,1,0)+z(0,0,1).

Pregazzix2a · Accepted Answer

In general we need to write down the system&amp;nbsp;av1+bv2+cv3=(x,y,z)&amp;nbsp;in augmented matrix form and handle it by RREF to find the condition on x,y,z for the system to have solution.&amp;nbsp;This approach provides the cartesian equations of the subspace spanned by&amp;nbsp;v1,v2,v3.&amp;nbsp;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.

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

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