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Showing 11 pages in Linear Algebra.
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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.
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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.
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Proof that the composition of linear maps is linear
Composing two linear maps does not break linearity. The proof is a careful unpacking of definitions: first pass through the inner map, then use linearity again in the outer map.
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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.
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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.
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Carry out Gram-Schmidt orthogonalization on a basis in
Gram-Schmidt turns a useful but tilted basis into an orthogonal one. The example keeps the arithmetic small enough that the projection idea remains visible.
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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.
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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.
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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.
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How a linear transformation changes the plane
A linear transformation of the plane is completely determined by where it sends the two standard basis vectors. From there, every grid point follows by the same linear combination.
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What a determinant measures geometrically
The determinant of a 2 by 2 matrix is the signed area scale factor of the transformation. It tells whether the unit square stretches, flips, or collapses.
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