Answer Happy

1. (Linear Maps: Kernels, Ranges, Rank-Nultity Theorem). Consider the linear map -21 +72-3y -274 + 21 3a -112 +4 +34 from the vector space R into the vector space R³. Transform the augmented matrix 7 -3 -20 5 2 1 0-10 -4 -11 0 4 3 3 0 -11 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 (1.1)); (ii) a basis for ker(L) and then the nullity mul(L) of L (iii) the rank rank (L) of L; 5 (iv) whether the vector & -10- is in the range L(R) of L (in other words, you have to determine whether the said linear system -11 associated with the matrix in (4.1) is consistent.) Present your answers to the problem in a talde of the following form: ). ( ER¹
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): ₁..., Hence nul(L) dim(ker(L))...; (iii) By (ii) and by the Rank-Nullity Theorem, rank(L) = ...; (iv) be L(R), or be L(R¹) (whichever is true)