Final answer:
To determine if vector b is in the column space of matrix a, set up the equation Ax = b. If a solution exists, b is in the column space and can be expressed as a linear combination of a's columns. The solution x provides the coefficients for the combination.
Step-by-step explanation:
The question is asking to determine whether a vector b is in the column space of a matrix a, and if so, to express b as a linear combination of the columns of a. This task involves concepts from linear algebra, particularly dealing with vector spaces and linear combinations.
To determine if b is in the column space of a, we can set up a matrix equation Ax = b, where x is a vector of constants. If this equation has a solution, then b is in the column space, and the solution provides the coefficients of the linear combination. If there's no solution, b is not in the column space.
If b is in the column space, and assuming that the columns of a are â₁, â₂, ..., âₙ, then b can be expressed as b = x₁â₁ + x₂â₂ + ... + xₙâₙ, where x₁, x₂, ..., xₙ are the components of vector x.