Questions: Systems of Linear Equations and Matrix Form

Question 1

A system Ax = b is consistent and A has free variables (rank less than the number of unknowns). What does the solution set look like?

• A. Exactly one solution
• B. No solutions
• C. Infinitely many solutions forming an affine subspace
• D. Two solutions

Answer: (C)

When the system is consistent (b is in the column space of A) and A has free variables, the null space of A is nontrivial. Every solution is a particular solution plus any vector from the null space, forming an affine subspace — a translated copy of the null space.

Question 2

A system Ax = b has a solution if and only if the rank of the augmented matrix [A | b] equals the rank of A.

• T. True
• F. False

Answer: True

This is the Rouché–Capelli consistency criterion. If rank([A | b]) > rank(A), then appending b adds a new pivot — meaning b is linearly independent from the columns of A and therefore outside the column space. In that case no combination of A's columns can equal b, so no solution exists.

Question 3

What is the geometric meaning of 'b is in the column space of A' in the equation Ax = b?

Model answer: b can be written as a linear combination of the columns of A, with the coefficients being the entries of the solution vector x. Geometrically, b lies within the subspace spanned by A's columns.

The columns of A are vectors in R^m. Ax computes a linear combination of those columns, so the set of all reachable right-hand sides is exactly the column space. If b lies in this span, a solution vector x exists; if not, no x can produce b under the mapping A.