MATH1961 · Mathematics 1a (advanced)
Linear Algebra
This is the structural half of MATH1961 (the Brownlowe stream). A vector in ℝn is an ordered n-tuple; two operations — componentwise addition and scalar multiplication — build everything, and the dot product injects geometry (length, angle, perpendicularity) into pure algebra. The first big proof is Cauchy–Schwarz via the discriminant trick (a single non-negative quadratic), which is what makes the cosine formula well-defined. From there: lines and planes in ℝ³ with the cross product, Gaussian elimination for linear systems, matrix algebra and inverses, and determinants. The abstract core is vector spaces — subspaces, span, linear independence, basis and dimension — culminating in the rank–nullity theorem, stated and proved. AHA-units throughout: display the theorem, give the proof steps, work an example, flag the trap.
What this chapter covers
- 01Vectors, the dot product & norm; Cauchy–Schwarz by the discriminant trick
- 02The cross product, lines & planes in ℝ³
- 03Gaussian elimination & solving linear systems
- 04Matrix algebra, inverses & determinants
- 05Vector spaces: subspace, span, independence, basis, dimension — and rank–nullity
Worked example: prove Cauchy–Schwarz by the discriminant trick
- +1Build a non-negative quadratic. For real t, let p(t) = (tu + v)·(tu + v). Since w·w ≥ 0 for any vector w, we have p(t) ≥ 0 for all t.
- +1Expand by the dot-product axioms. p(t) = ‖u‖² t² + 2(u·v) t + ‖v‖² — a real quadratic in t with positive leading coefficient (assume u ≠ 0; if u = 0 the inequality is trivial).
- +1Use the discriminant. A quadratic at² + bt + c with a > 0 stays ≥ 0 for all t iff its discriminant b² − 4ac ≤ 0.
- +1Read off the inequality. Here b² − 4ac = 4(u·v)² − 4‖u‖²‖v‖² ≤ 0, i.e. (u·v)² ≤ ‖u‖²‖v‖².
- +1Take square roots. |u·v| ≤ ‖u‖‖v‖, with equality iff p has a real root iff u and v are parallel. ■
Key terms
- Dot product
- u·v = u1v1 + … + unvn, a scalar. It defines length ‖v‖ = √(v·v) and orthogonality (u ⊥ v iff u·v = 0), and the geometric formula u·v = ‖u‖‖v‖cosθ gives the angle between vectors.
- Cauchy–Schwarz inequality
- |u·v| ≤ ‖u‖‖v‖ for all vectors u, v, with equality iff they are parallel. It is what makes the cosine ratio (u·v)/(‖u‖‖v‖) sit in [−1, 1], and it is proved from a single non-negative quadratic (the discriminant trick).
- Gaussian elimination
- The systematic row-reduction of a system’s augmented matrix to row-echelon form, from which the solution set is read off. It classifies a system as having a unique solution, infinitely many (free variables), or none (an inconsistent row).
- Basis and dimension
- A basis of a vector space is a linearly independent set that spans it; every basis of a given space has the same number of vectors, which is the dimension. A basis lets every vector be written uniquely as a linear combination of the basis vectors.
- Rank–nullity theorem
- For a linear map (or matrix) with domain of dimension n, rank + nullity = n: the dimension of the image plus the dimension of the kernel equals the number of input dimensions. It links the solvability of Ax = b to the structure of A and is proved by extending a basis of the kernel.
Linear Algebra FAQ
What is the discriminant trick and why does the exam love it?
It proves the Cauchy–Schwarz inequality from a single observation: the squared length p(t) = (tu + v)·(tu + v) is ≥ 0 for every real t, so this quadratic in t never dips below the axis, which forces its discriminant ≤ 0 — and that discriminant is Cauchy–Schwarz. The exam likes it because the marks are explicitly in saying why p(t) ≥ 0 (a dot square) and why the discriminant is ≤ 0; quoting the inequality without the argument scores little.
How do I tell whether a linear system has none, one, or infinitely many solutions?
Row-reduce the augmented matrix by Gaussian elimination. A pivot row like [0 0 … 0 | nonzero] means no solution (inconsistent). Otherwise, if every variable is a pivot (leading) variable there is a unique solution; if some variables are free (non-pivot columns) there are infinitely many, parameterised by those free variables.
What is the difference between span, linear independence and basis?
Span is the set of all linear combinations of some vectors (everything they can reach). Linear independence means none of them is a redundant combination of the others. A basis is the sweet spot: independent and spanning, so it reaches the whole space with no redundancy — and every basis of a space has the same size, the dimension.
What does the rank–nullity theorem actually tell me?
That for a matrix A acting on n-dimensional inputs, rank(A) + nullity(A) = n — the dimensions of the image and the kernel always add up to the number of columns. It is the bookkeeping behind ‘more unknowns than independent equations means free variables’, and the Advanced unit asks you to prove it (extend a basis of the kernel to a basis of the domain).
Exam move
Split the chapter into computation and structure. For computation — dot/cross products, Gaussian elimination, inverses, determinants — drill the algorithms until they are fast and accurate by hand, since arithmetic slips cost easy marks. For structure — Cauchy–Schwarz, vector-space axioms, span/independence/basis, rank–nullity — learn the proofs, because that is the Advanced premium. Rehearse the Cauchy–Schwarz discriminant proof until you can write it cleanly, stating why at each step. For vector spaces, practise verifying the subspace conditions and writing independence as ‘the only combination giving 0 is the trivial one’.
