Find X4 to Make {X1, X2, X3, X4} Linearly Independent

[mccoy1]

Homework Statement

Hi fellows,
If we are given 3 vectors (e.g X1, X2, X3) in R^4, how would we find X4 such
that {X1, X2, X3, X4} is a linearly independent set?

Homework Equations

The Attempt at a Solution

I tried something like this: aX1 + bX2 + cX3 + dX4 =0, but it didn't work.
Some tips would be highly appreciated.
Cheers.

 

[Mark44]

Homework Statement

Hi fellows,
If we are given 3 vectors (e.g X1, X2, X3) in R^4, how would we find X4 such
that {X1, X2, X3, X4} is a linearly independent set?

Homework Equations

The Attempt at a Solution

I tried something like this: aX1 + bX2 + cX3 + dX4 =0, but it didn't work.
Some tips would be highly appreciated.
Cheers.

Since your vectors x₁, x₂, and x₃ are linearly independent, they span a three-dimensional subspace of R⁴. This subspace is represented by {c₁x₁ + c₂x₂ + c₃x₃}, where c₁, c₂, and c₃ are arbitrary scalars. A vector x₄ that is not in the span of the given three vectors satisifies the equation
x₄ \cdot (c₁x₁ + c₂x₂ + c₃x₃) = 0.

Presumably from this equation you can find the coordinates of x₄ to get the fourth vector of your basis. Note that there will be many vectors that work, but all of them have the same direction, meaning that all are scalar multiples of each other.

 

[mccoy1]

A vector x₄ that is not in the span of the given three vectors satisifies the equation
x₄ \cdot (c₁x₁ + c₂x₂ + c₃x₃) = 0.

Presumably from this equation you can find the coordinates of x₄ to get the fourth vector of your basis.

Wow, thank you ver much Mark44. I'll try to find other vectors using that relation. I got the correct answer two days ago but it was through trial and error, which take a very long time.