Conditions for Vector Membership in a Spanned Vector Space

In linear algebra, determining whether a vector belongs to a spanned vector space is a fundamental task. This article explores the conditions required for a vector ( mathbf{u} mathbf{v} mathbf{w} ) to be a linear combination of two given vectors and thus a member of the spanned vector space. The vectors in question are ( mathbf{1} mathbf{2} mathbf{3} ) and ( mathbf{-1} mathbf{2} mathbf{4} ).

Introduction to the Problem

The given vectors are:

( mathbf{1} mathbf{2} mathbf{3} ) ( mathbf{-1} mathbf{2} mathbf{4} )

Any vector ( mathbf{u} mathbf{v} mathbf{w} ) must be a linear combination of these vectors to belong to the spanned vector space. This means that:

[ mathbf{u} mathbf{v} mathbf{w} a mathbf{1} mathbf{2} mathbf{3} b mathbf{-1} mathbf{2} mathbf{4} ]

Formulating the Linear Combination

Expressing ( mathbf{u} mathbf{v} mathbf{w} ) in terms of the given vectors, we get:

[ a mathbf{1} mathbf{2} mathbf{3} b mathbf{-1} mathbf{2} mathbf{4} mathbf{u} mathbf{v} mathbf{w} ]

This results in the following system of equations:

[ u a - b ] [ v 2a 2b ] [ w 3a 4b ]

Solving for Scalars a and b

To solve for the scalars ( a ) and ( b ), we first isolate ( b ) from the first equation:

[ b a - u ]

Substituting ( b ) into the second equation gives:

[ v 2a 2(a - u) ] [ v 4a - 2u ]

From this, we can express ( a ) as:

[ a frac{v 2u}{4} ]

Next, substitute ( a ) back into the expression for ( b ):

[ b frac{v 2u}{4} - u ][ b frac{v 2u - 4u}{4} ][ b frac{v - 2u}{4} ]

Substituting a and b into the Third Equation

Now substitute ( a ) and ( b ) into the equation for ( w ):

[ w 3a 4b ] [ w 3left(frac{v 2u}{4}right) 4left(frac{v - 2u}{4}right) ]

Simplifying this:

[ w frac{3(v 2u) 4(v - 2u)}{4} ][ w frac{3v 6u 4v - 8u}{4} ][ w frac{7v - 2u}{4} ]

Therefore, the condition for ( mathbf{u} mathbf{v} mathbf{w} ) to be in the span of the vectors ( mathbf{1} mathbf{2} mathbf{3} ) and ( mathbf{-1} mathbf{2} mathbf{4} ) is:

[ 4w 7v - 2u ]

Rearranging gives the final condition:

[ 2u 4w 7v ]

Conclusion

To summarize, the vector ( mathbf{u} mathbf{v} mathbf{w} ) belongs to the space spanned by the vectors ( mathbf{1} mathbf{2} mathbf{3} ) and ( mathbf{-1} mathbf{2} mathbf{4} ) if and only if the following condition holds: