Math 251 — Each question gives you a real problem. Pick the correct method to solve it.
How to check sizes, find a specific entry, compute products & scalar operations · 12 questions
How to find direction vectors, normal vectors, parametric & general forms · 10 questions
How to row-reduce, spot contradictions, count free variables, read solutions · 12 questions
How to set up and interpret span tests & independence tests · 12 questions
See a problem cold — choose the right method — no warm-up · 8 questions
Exam-style problems to work by hand on paper. Show every row operation.
Write the augmented matrix, apply row ops labelling each as R₂ → R₂ − 2R₁, reach RREF, classify the solution (unique / ∞ many / inconsistent), write the general solution.
Compute u = P₂−P₁ and v = P₃−P₁, find n by solving n·u=0 and n·v=0, write ax+by+cz=d, verify all 3 points satisfy it.
Form [v₁|v₂|v₃], row-reduce to RREF. No free variables → independent. Free variable → dependent; find the dependence relation by setting the free parameter to 1.
Row-reduce [v₁|v₂|b]. If consistent → read off c₁ and c₂ and write b = c₁v₁ + c₂v₂. If inconsistent → b is NOT in the span.