MA 35100 Lecture Notes - Lecture 24: Linear Map, If And Only If
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If you have a samatrix (nxn), we can multiply it by itself as many times as you want. So a is a zero/root of al-8a+ 11 =0. Note:ca. any pl a, b, +acbsanbu =(b. bu] 1)" swapping the order role (column/row) gives the same answer. The rank of a matrix: #linindpcol:#lin indep rows a=(ii) b:/ 2. Any linear dependencies amongst the columns of bare kept amongst the columns of ab (by lin properties). Ar. it comparing rows of a with rows of ab. same idea from above applies when rank (ab) =#lin indep rows of ab = #lin indeprows of a = rank (al. Theorem 3. 3: if ab is defined, rank (ab) = rank (a) rank (ab) 1 rank (b) For any linear transformations, t. i= w and is:tv - u. ), the composition of is with t, is the transfrth 720t =5x7 given by (t2ot. If t: t- t is a transfrth, then t is invertible ifthe equation.
