Chapter 0 wrap-up

Readiness review

Closing thought

Every idea in this chapter points at one stubborn matrix. The quarter-turn rotation of the plane moves every nonzero vector off its own line, so it has no real eigenvector. Its characteristic polynomial is $\lambda^2+1$ , which has no real roots. The old toolkit of basis, rank, similarity and eigenvalues still applies to it. It needs a larger number system to finish the job.

You can now factor that polynomial over $\mathbb C$ as $(\lambda-i)(\lambda+i)$ and read off the conjugate pair $\pm i$ . You can also tell when a real polynomial must keep a quadratic factor over $\mathbb R$ but splits completely over $\mathbb C$ . Chapter 1 asks a question these numbers raise. Does a vector space need real scalars at all? The answer is no: any number system where you can add, subtract, multiply and divide will do.

Recipes in this chapter

Step-by-step procedures for the problem types this chapter teaches.

Compute a complex inner product

\langle u, v\rangle

Complex numbers and complex arithmetic you'll need Factor a polynomial completely and state each root's multiplicity Polynomials: division, roots, and factorization over ℝ and ℂ

Chapter review

One practice set per topic this chapter taught — same fixed, ungraded ladders as the practice panels inside each section.

Practice: Complex numbers and polynomials (2 problems · doesn’t affect your progress)

Given $z = a+bi$ , the complex conjugate $\overline z$ is obtained by

Correct! 🎉 Not quite.

Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.

That negates z entirely (-z), not the conjugate — the real part of a conjugate never changes.

The real part stays exactly the same under conjugation; only the imaginary part's sign flips.

Swapping the parts is a different operation entirely and isn't what conjugation means.

Why

$\overline{a+bi} = a-bi$ — only the imaginary part's sign changes.

Flipping the sign of the imaginary part only

The multiplicity $k$ of a root $r$ is defined as

Correct! 🎉 Not quite.

Term count has nothing to do with how many times a particular root repeats.

Multiplicity describes a ROOT's repetition, not how a coefficient happens to look.

Degree is the polynomial's overall size — a degree-5 polynomial can have a root of multiplicity 1, 2, 3, 4, or 5.

Why

Multiplicity is exactly how many times $(x-r)$ divides out before a nonzero remainder appears.

$(x-r)^k$ divides evenly but $(x-r)^{k+1}$ does not

Take the chapter quiz On to Chapter 1 →

← Polynomials: division, roots, and factorization over ℝ and ℂ Vector spaces over a field →