mccoy1 said: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 said:A vector x4 that is not in the span of the given three vectors satisifies the equation
x4 [itex]\cdot[/itex] (c1x1 + c2x2 + c3x3) = 0.
Presumably from this equation you can find the coordinates of x4 to get the fourth vector of your basis.
Linear independence refers to the property of a set of vectors where none of the vectors can be expressed as a linear combination of the other vectors in the set. In other words, no vector in the set can be written as a sum of scalar multiples of the other vectors in the set.
To determine if a set of vectors {X1, X2, X3, X4} is linearly independent, you can use the matrix method. Construct a matrix A with the vectors as its columns. Then, solve the equation Ax = 0, where x is a vector of coefficients. If the only solution is x = 0, then the set of vectors is linearly independent. Another method is to use the determinant of the matrix A. If det(A) is not equal to 0, then the set of vectors is linearly independent.
The purpose of finding X4 is to determine the minimum number of vectors needed to form a linearly independent set. This is useful in many mathematical and scientific applications, such as solving systems of equations, finding basis vectors, and determining whether a set of vectors spans a given space.
No, a set of vectors cannot be linearly independent in one dimensional space. In one dimensional space, all vectors lie on the same line and can be expressed as scalar multiples of each other. Therefore, any set of vectors in one dimensional space will be linearly dependent.
The addition of X4 can make {X1, X2, X3, X4} linearly independent if X4 is not a linear combination of X1, X2, and X3. If X4 can be expressed as a linear combination of X1, X2, and X3, then the set will remain linearly dependent. In other words, adding a new vector can only increase the linear independence of a set, but it may not necessarily make the set linearly independent.