Textbook Readings Unit Four Matrices in Engineering 8 1 Engineering problems produce symmetric matrices K K is often positive de nite K is the stiffness matrix and K is the structure s response to the forces from the outside F Kx equilibrium force balance F is the gravity of masses K is the stiffness matrix and x is the downward displacement Fixed Fixed Fixed Free Free Free Circular The spring constant c for these matrices is one K is given in the form K A CA and in this case C I Movement of the masses u u u When a mass moves downward its displacement is positive Tensions in the springs y y y Elongations of springs e e e External forces on the masses f f f Linear Transformations 7 1 7 2 When a matrix A multiplies a vector v it transforms v into another vector Av Matrix multiplication T v Av gives a linear transformation A transformation T assigns an output T v to each input vector v in V The transformation is linear if it meets these requirements for all v and w T v w T v T w T cv cT v for all c All together we get T cv dw must equal cT v dT w Shift is not linear v w u is not T v T w v u w u The exception is when u 0 Example a 1 3 4 and let T v be the dot product a v The input is v v v v and the output is T v v 3v 4v The inputs v come from three dimensional space so V R The outputs are just numbers so the output space is W R We are multiplying by the row matrix A 1 3 4 so T v Av Review of the Key Ideas A transformation T takes each v in the input space to T v in the output space T is linear if T v w T v T w and T cv cT v lines to lines Combinations to combinations T c v c v c T v c T v The transformation T v Av v is linear only if v 0 then T v Av Choice of Basis 7 3 Linear Transformation T c v c v c T v c T v If we know know T v T v and if v v a basis for the input means that every vector is one combination of the v s If we know T v for v basis vectors v v Then we know T v for all v Next When is T invertible Know that T v w where j 1 n w w must be a basis for the output space A v v w w AV W when is A invertible We know V is invertible A WV therefore A is invertible when W is invertible Dimension of the input space is equal to the dimension of the output space which is the same as the column space and the dimension of the null space Every linear transformation T can be described as a matrix T v x v x Example input space v x a a x a x basis 1 x x output space apply basis T 1 0 T x x T x 2x Notice that this is like eigenvectors with eigenvalues 0 1 2 matrix this is the same as thus T v x v x 0 a x 2a x Columns of W are the new basis x Wc where x is the old matrix Review of the Key Ideas If we know T v T v for a basis linearity will determine all other T v The derivative and integral matrices are one sided inverses constant x 0 Derivative Integral I is the Fundamental Theorem of Calculus If A and B represent T and S and the output basis for S is the input basis for T then the matrix AB represents the transformation T S u The change of basis matrix M represents T v v Its columns are the coef cients of the output basis expressed in the input basis w m v m v
View Full Document
Unlocking...