Class Note for MATH 410 at UA
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Date Created: 02/06/15
THE UNIVERSITY Q OF ARIZONA Math 410 Matrix Analysis Section 25 The Fundamental Matrix Subspaces Section 26 Graphs and Incidence Matrices Paul Dostert January 19 2009 A Range and Kernel The range of an m gtlt n matrix A is the subspace mgA C Rm spanned by its columns The range is also called the column space or image of the matrix Note that we can write mgA Aac1r E R C Rm which means that b e rngA iff Ax b has a solution there is some c so that Ax b The kernel of A is the subspace kerA C Rquot consisting of all vectors which are annihilated by A so kerA z e RnAz 0 C R The kernel is often reffered to as the null space The kernel is simply the set of solutions to Az 0 Thm If zl zk are solutions to the same homogenous linear system Az 0 then so is any linear combinaton clzl Ckzk Ex Prove this A Examples of Kernel and Range 1 1 1 Ex Compute the mg and ker of span 2 1 2 0 2 8 Thm The linear system Ax b has a solution x iff b e rngA If this occurs then x is a solution to the linear system iff c c z with z e kerA Recall ln DEs if have a particular solution 90 then the general solution is the particular solution plus the solution to the homogenous problem Prop If A is an m gtlt n matrix then TFSAE i kerA 0 ii rankA n iii Ax b has no free variables iv Ax b has a unique solution for each I e rngA Prop If A is a square n gtlt n matrix then TFSAE i kerA 0 ii rankA n iii A is nonsingular iv mgA R As The Superposition Principle Thm Suppose that we know particular solns x31 systems 796 to each of the linear 14331917 14331927 Axbk with A and c the same in each eqn and bi e rngA7 z 17 7 k Then for any choice of scalars 017 7 ck a particular soln to the combined system AC1blCkbk is the superposition x01x Ckx of individual soln The general soln to the combined system is c c A where z e kerA is the general soln to Az 0 The main point of this theorem is that we can write solns to Ax b in terms of linear combinations of different bases 80 if we write bb1 1bm m then solve Axl 62 for each i then we can find a soln to Axquot b by 30 9136 er bmxfn Ala Superposition Examples Ex Express the solution to Axltazlgtltgtlt2gtb as a linear combination of the solutions to A301 el and A902 e2 Ex Applying a unit force in the horizontal direction makes as mass move 2 units to the right while unit force in the vertical direction makes the mass move 1 unit up Assuming linearity where will the mass move under a force of f 173 A Adjoint Cokernel and Corange The adjoint to a linear system Ax b of m equations in n unknowns is the linear system ATy f consisting of n equations in m unknowns y e Rm with rhs f e Rquot The corange of an m gtlt n matrix A is the range of its transpose corngA rngAT ATyy E Rm C R The cokernel or left null space of A is the kernel of its transpose cokerA kerAT w e RmATw 0 c Rm Fundamental Thm of Linear Algebra Let A be an m gtlt n matrix of rank 7quot Then dim corngA dim mgA rankA rankAT 7quot dim kerA n r7 dim cokerA m 7quot A Adjoint Cokernel and Corange To actually compute the range kernel corange and cokernel of an m gtlt n matrix A with ref U we use the following process 0 mg A choose the 7quot columns of A where the pivots appear in U o ker A write the general soln to Ax 0 as a linear combo of the n 7quot basis vectors whose coefficients are the free variables corng A choose the 7quot nonzero rows of U coker A write the general soln to ATy 0 as a linear combo of the m 7quot basis vectors whose coeffs are the free variables Ex Find the range corange kernel and cokernel of 1 1 a A0 4 1 1 2 b 2 1 3 3 1 4 A Directed Graphs and Incidence Matrices We will delay the discussion of this topic until later in the semester If you are interested in graph theory I would suggest you look into this topic a bit on your own Once we learn a little about Markov Chains we can use them in addition with directed graphs to study many interesting topics such as how a search engine does page rankings Ah Matlab Recall that in previous notes we found a basis for the column space of 1 1 0 1 0 1 1 1 0 1 1 1 using the rref We also have a function null which returns a normalized orthonormal actually basis for the null space of A A1 1 0101 1 101 1 1 nullA nullA gives us 057747 057747 057747 00000 as a basis for the null space which can be rescaled to 17 17 17 0 Clearly to find the corange or cokernel we can apply the same operations on the transpose we make a AT variable since A jb doesn t work in some versions of Matlab A1 1 0101 1 101 1 1 ATA Rjb rrefAT ATjb null AT
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