Final answer:
To create a system of two equations in four unknowns spanned by x and y, we express an arbitrary vector in the span as a linear combination of x and y, then choose two equations from this system to represent the relationship among the unknowns.
Step-by-step explanation:
To find a system of two equations in four unknowns whose solution set is spanned by x and y, we can first recognize that any vector in the span of x and y can be written as a linear combination c₁x + c₂y. If we denote an arbitrary vector in this span as [a,b,c,d]t, where a, b, c, and d are the unknowns, we can set up the following system of equations:
a = c₁*1 + c₂*1
b = c₁*0 + c₂*(-1)
c = c₁*2 + c₂*0
d = c₁*0 + c₂*2
By choosing two equations from this system that involve all four unknowns and only two arbitrary coefficients (c₁ and c₂), we can arrive at our desired system. For example, selecting the first and third equations, we get:
These two equations represent the relationships among the four unknowns that are consistent with any vector being a linear combination of x and y.