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Fill in the blanks about a linear map T:V→WT:V\to W between finite-dimensional vector spaces.

The rank–nullity theorem says dim⁡(V)=rank⁡(T)+nullity⁡(T)\dim(V)=\operatorname{rank}(T)+\operatorname{nullity}(T). So if dim⁡(V)=10\dim(V)=10 and rank⁡(T)=7\operatorname{rank}(T)=7, then nullity⁡(T)=______.Thismeansthesolutionspaceof\operatorname{nullity}(T)=\_\_\_\_\_\_. This means the solution space of T(\mathbf{v})=\mathbf{0}hasdimension______.Inparticular,thereare______linearlyindependentvectorsinhas dimension \_\_\_\_\_\_. In particular, there are \_\_\_\_\_\_ linearly independent vectors in\ker(T)$$.

Course
Introductory Non‑Relativistic Quantum Mechanics: Postulates, Ope
12 units57 lessons
Topics
Quantum Physics (Non-relativistic Quantum Mechanics)Mathematical PhysicsLinear AlgebraDifferential Equations / Boundary-Value ProblemsComplex Analysis (foundational tools)
About this course

Develop an intuition-first, problem-solving mastery of non-relativistic quantum mechanics using the postulates and operator formalism. Core topics include states and representations (kets/bras and wavefunctions), normalization and phase, the Born rule and expectation values, measurement as projectors and spectral decomposition, and unitary time evolution via the Schrödinger equation and U(t)=e^{-iHt/ħ}. Apply commutators and uncertainty relations, switch between position and momentum pictures, and solve standard Hamiltonians: 1D wells and barriers (including tunneling and probability current), the harmonic oscillator (ladder operators), angular momentum and spin-1/2, and the hydrogen atom. Gain moderate facility with approximation methods such as time-independent perturbation theory, the variational principle, and optional WKB.