is an upper triangular matrix tracking the geometric adjustments. 5. Determinants and Eigenvalues
The space spanned by all linear combinations of the rows of (or columns of ATcap A to the cap T-th power Dimension: Location: A subspace of 4. The Left Nullspace Definition: The set of all vectors Dimension: Location: A subspace of The Fundamental Theorem of Linear Algebra
linearly independent eigenvectors, we can stack them into the columns of an eigenvector matrix . This allows us to diagonalize
systems. He introduces the (intersecting lines) and the Column Picture (combining vectors). Understanding the Column Picture is the "aha!" moment for most students. 2. Matrix Multiplication and Factorization
[ \det(A - \lambda I) = 0 ] This yields (n) eigenvalues (counting multiplicities).
), each equation represents a flat plane in three-dimensional space ( R3cap R cubed