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9.3 Gram-Schmidt orthogonalization
Apply Gram-Schmidt to turn a basis into an orthogonal or orthonormal basis while preserving the same span.
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MATH1030: Linear algebra I
Rigorous linear algebra notes on systems, matrices, structure, and proof, with interaction used only where it clarifies the mathematics.
Chapter 1
Systems of equations
Learn to read equations as full solution sets.
Chapter 2
Matrices and elimination
Build matrix intuition and use row reduction with purpose.
Chapter 3
Matrix algebra
Matrix multiplication, transpose, and structural matrix notation.
Chapter 4
Solution structure
Homogeneous systems, null spaces, and the shape of full solution sets.
Chapter 5
Invertibility
Understand when a matrix can be undone and why that matters.
Chapter 6
Vector spaces
Move from matrix procedures to the structure of spaces, span, independence, and basis.
Chapter 7
Determinants
Determinants, cofactor formulas, and the structural algebra that connects row operations, transpose, and invertibility.
Chapter 8
Eigenvalues and diagonalization
Eigenvalues, eigenspaces, similarity, and diagonalization as the next structural layer after determinants.
Chapter 9
Inner products and orthogonality
Inner products, orthogonality, orthonormal bases, and Gram-Schmidt as the geometric layer after eigenvalues.