Learn Math with live MCPCalc tools
Explore proofs, worked examples, and clear explanations, then try the math yourself with graphing, CAS, proof builder, spreadsheets, and calculators.
Showing 7 pages in Linear Algebra.
Filter By Subject
Proofs
Structured arguments in Proof Builder, with theorem statements, line-by-line reasoning, and nearby variations.
Worked Examples
Standard problems where the computation and the explanation stay tied to the live tool.
Explanations
Concept pages that build intuition first and then connect it to formal notation and exact calculations.
Proofs
A catalog of interesting and common proofs to learn, study, and revisit.
Proof that the kernel of a linear map is a subspace
This is a standard linear algebra proof because it packages the subspace test into one reusable pattern: show zero is inside, then check closure under addition and scalar multiplication.
Open proof →
Proof that eigenvectors for distinct eigenvalues are linearly independent
Distinct eigenvalues do more than label eigenspaces. They force eigenvectors to be independent, which is exactly why diagonalization becomes possible so often in practice.
Open proof →
Worked Examples
Common examples from different branches of math, each written with clear step-by-step instructions.
Diagonalize a matrix and compute
Diagonalization is one of the first places students see why eigenvalues matter computationally. Once , powers of become powers of a diagonal matrix.
Open worked example →
Compute eigenvalues and eigenvectors of a matrix
A first serious 3 by 3 eigenproblem should show new structure without burying the point in arithmetic. This one does that: you see the characteristic polynomial, the eigenspaces, and the diagonalization test in one pass.
Open worked example →
Explanations
Concept pages that build intuition first and then connect it to formal notation and MCPCalc tools.
What eigenvectors and eigenvalues mean geometrically
An eigenvector is a direction that survives a linear transformation without rotating away from its own line. The eigenvalue tells you the scale and possible sign flip along that direction.
Open explanation →
What linear independence means geometrically
Linear independence is the condition that nothing in the list is wasted. Geometrically, each new vector must add a new direction instead of repeating one you already had.
Open explanation →
What a basis does in linear algebra
A basis is what turns a vector space into something you can navigate. It reaches every vector, and it does so without redundancy, so coordinates become possible.
Open explanation →