1 Linear Maps Kernels Ranges Rank Nullity Theorem Consider The Linear Map 21 22 23 101 3 1 L 1422 1022 21 22 1 (39.21 KiB) Viewed 34 times
1. (Linear Maps: Kernels, Ranges, Rank-Nullity Theorem). Consider the linear map 21 22 23 -101 3.1 L +1422 -1022 +21.22 --733 +6013 - 1123 -34 +234 ER -621 13 from the vector space R into the vector space R. Transform the augmented matrix () -4 3 -6 14 - 10 21 -7 6 -11 -30 2 0 -4 0 6 0 (4.1) 7 to reduced row-echelon form (having two columns of constant terms added to the matrix of the above linear system, we can simultaneously solve two linear systems), and then determine: (i) the kernel ker(L) of L (as the general solution of the first linear system associated with the matrix in (4.1)); (ii) a basis for ker(L) and then the nullity nul(L) of L; (iii) the rank rank(L) of L: - 0 (iv) whether the vector 6 = 0 is in the range L(R) of L (in other words, you have to determine whether the second linear system associated with the matrix in (4.1) is consistent.) Present your answers to the problem in a table of the following form:
Present your answers to the problem in a table of the following form Subproblem | Answer(s) (i) ) The general solution of the first (homogeneous) linear system associated with the augmented matrix (4.1): (ii) A list/set of vectors that form a basis for ker(L): c = ...... Hence nul(L) = dim( ker(L)) = ...; (iii) By (ii) and by the Rank-N ullity Theorem, rank(L) = ...; (iv) ) beL(R), or 6 L(R) (whichever is true)
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