Basically, you need to understand the abstract properties of Linear Algebra, e.g. group theoretic properties, etc. This is in contrast to "undergraduate" Linear Algebra, which focuses primarily on computational aspects and some basic algebraic properties (e.g. rank-nullity theorem, etc.).
For graduate-level multivariable calculus, you need to understand rigorous proofs regarding integration and differentiation in $\Bbb R^n$, as well as analytic properties of differential forms. This differs from undergraduate multivariable calculus, which again is typically computational, and focuses on vector calculus and use of Green's/Stoke's Theorems, rather than their construction and proof.