Final answer:
To determine if 'u' is in the plane spanned by the columns of 'a', check if 'u' can be expressed as a linear combination of the columns of 'a'.
Step-by-step explanation:
To determine whether 'u' is in the plane spanned by the columns of 'a', we need to check if 'u' can be expressed as a linear combination of the columns of 'a'.
If 'u' can be written as 'u = c1*a1 + c2*a2 + ... + cn*an' where 'a1', 'a2', ..., 'an' are the columns of 'a' and 'c1', 'c2', ..., 'cn' are scalars, then 'u' is in the plane spanned by the columns of 'a'.
If 'u' cannot be written in this form, then 'u' is not in the plane spanned by the columns of 'a'.
To determine the values of 'c1', 'c2', ..., 'cn', we can solve the system of equations formed by setting 'u' equal to the linear combination of the columns of 'a'.
For example, if 'u' is a 3-dimensional vector and 'a' is a 3x3 matrix, we would solve the system of equations:
c1*a1 + c2*a2 + c3*a3 = u