How can I structure proofs in linear algebra beyond computations?

[Violet.B]

I'm a second-year math major taking my first rigorous linear algebra course, and I'm hitting a wall with proofs. I understand the computational aspects, but when asked to prove something like why the set of all polynomials of degree less than or equal to n is a vector space, I struggle to structure my arguments clearly. I know I need to verify the axioms, but my proofs feel messy and I often overlook a subtle closure property. Are there any specific proof techniques or mental frameworks you found helpful when making the jump from calculation to proof in this subject? I'd appreciate any advice on organizing my thoughts.

[Brandon.R]

Proofs click for me when I treat them like a checklist. For the set of polynomials with degree ≤ n, I start by clearly stating the object, then verify the subspace criteria: it contains the zero polynomial, and it’s closed under addition and scalar multiplication. The tricky part is rarely the operations themselves—the sum of two degree ≤ n polynomials also has degree ≤ n, and scaling doesn't raise the degree.

[JohnXM]

Here's a small skeleton you can use: let p(x)=a0+...+an x^n and q(x)=b0+...+bn x^n. Then (p+q)(x)= (a0+b0)+...+(an+bn)x^n, so deg(p+q) ≤ n. For any scalar c, deg(cp) ≤ n as well. So P≤n is a subspace of R[x] with the usual operations.

[Sofia_W]

Another handy frame is the coefficient-vector view. Map p to (a0,...,an). Addition and scalar multiplication go coordinate-wise, so closure is immediate. This also shows the dimension is n+1. If you want, you can make it formal by defining φ: P≤n → R^{n+1} by φ(p)=(a0,...,an) and noting φ is a linear bijection.

[Ryan.W]

I sometimes prefer reframing: you can prove it's a vector space by showing it's isomorphic to a known space, rather than grinding through axioms. Just be careful to justify the map and its inverse, not pretend it’s obvious.

[LarryL]

Depends on how formal your course expects you to be. Want a ready-to-use write-up? I can draft a clean, bullet-proof version you can adapt for exams. Also, do you work in R[x] over the real numbers, or another field?

[EricLW]

Yep, I’ve tripped over the nitty-gritty before. A quick check list helps: 0 polynomial in the set? yes. Closure under + and scalar mult? verify with generic p and q. Small, precise steps beat a vague argument every time.