Lecture Notes For Linear Algebra Gilbert Strang !!install!! <EXTENDED · 2025>
: The space containing all solutions to the homogeneous equation . It resides in Rncap R to the n-th power and has dimension The space spanned by the rows of (columns of ATcap A to the cap T-th power ). It resides in Rncap R to the n-th power and has dimension Left Nullspace, : The nullspace of ATcap A to the cap T-th power , satisfying . It resides in Rmcap R to the m-th power and has dimension Orthogonality of the Subspaces The fundamental subspaces are perpendicular to each other: The Row Space is orthogonal to the Nullspace in Rncap R to the n-th power The Column Space is orthogonal to the Left Nullspace in Rmcap R to the m-th power 4. Orthogonality and Least Squares When a real-world system has more equations than variables (
?" If the three column vectors lie in the same flat plane, they cannot fill 3D space, and certain vectors will be unreachable (a singular matrix). 2. Matrix Elimination and Factorization ( lecture notes for linear algebra gilbert strang
A=(E21-1E31-1E32-1)U=LUcap A equals open paren cap E sub 21 to the negative 1 power cap E sub 31 to the negative 1 power cap E sub 32 to the negative 1 power close paren cap U equals cap L cap U Where : The space containing all solutions to the
