Let V1=(1,2,-5)^T v2=((2,5,6)^T and b=(7,4,-3)^T cn be expressed as a linear combination of V1 and V2?

I assume the question is whether the vector b is a linear combination of v₁ and v₂. If it is, then that means there are scalars c₁ and c₂ such that

b = c₁ v₁ + c₂ v₂   ==>   (7, 4, -3) = c₁ (1, 2, -5) + c₂ (2, 5, 6)

This gives the system of equations,

c₁ + 2c₂ = 7

2c₁ + 5c₂ = 4

-5c₁ + 6c₂ = -3

which has no solution, since

-2 (c₁ + 2c₂) + (2c₁ + 5c₂) = -2(7) + 4   ==>   c₂ = -10

but substituting this into any two equations in the system gives contradictory results. For instance,

c₁ + 2c₂ = c₁ - 20 = 7   ==>   c₁ = 27

2c₁ + 5c₂ = 2c₁ - 50 = 4   ==>   c₁ = 27

-5c₁ + 6c₂ = -5c₁ - 60 = -3   ==>   c₁ = -57/5

This means that b is not a linear combination of v₁ and v₂.

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