Lecture Notes For Linear Algebra Gilbert Strang |top|

. Its columns are the (eigenvectors of ATAcap A to the cap T-th power cap A ). They form an orthonormal basis for the input space The Geometric Meaning The SVD asserts that for any linear transformation: We can choose an orthonormal basis in the input space (

“The laws of nature are linear, and the rest are nonlinear approximations.” — Gilbert Strang lecture notes for linear algebra gilbert strang

The projection of (b) onto a vector (a) is: [ p = a\fraca^Tba^Ta = \fracaa^Ta^Ta b ] The projection matrix onto a line: (P = \fracaa^Ta^Ta). For a Every matrix, no matter how lopsided

For a

Every matrix, no matter how lopsided or messy, could be broken into three perfect pieces: a rotation, a stretching, and another rotation ( For a Every matrix

How much of the first column vector plus how much of the second column vector do we need to reach the vector [03]the 2 by 1 column matrix; 0, 3 end-matrix; yields the correct combination. Higher Dimensions ( and Beyond)