[Use ERO to find inverse](a) Use ERO to derive the formula for the inverse of the 2-by-2 matrixabcd. Whatassumptions do you need for the inverse to exist?(b) Repeat part (a) for the block partition matrixABCDwhereA, B, C, Dare ma-trices of compatible dimensions. What assumptions do you need for the inverse toexist?KVL/Jan19
EE2007/IM2007 Tutorial 4Linear AlgebraLinear Combination, Independence, Span, BasisExercise 19.
[Concepts and Examples discussed in lectures]The following were discuss in lectures.Review them and ask questions in the tutorialclass, if necessary. Generate your own example to test your understanding.Givenv,v1,v2andv3inR3. What do you have to do to(a) Determine whethervis a linear combination ofv1,v2andv3(b) Findspan(v1,v2) andspan(v1,v2,v3)(c) Determine whethervis inspan(v1,v2,v3)(d) Determine whetherv1,v2,v3are linearly independent or not.Exercise 20.[Linear Combinations](a) LetA=12-1234-321andx=-1-13.Write the productAxas a linearcombination of the column vectors ofA.(b) LetA=-1-234andB=3225.Write each column vector ofABas alinear combination of the column vectors ofA.(c) Describe all vectors inR3that can be written as a linear combination of the vectors12-1,37-2,130[ [a, b, c]0such that 3a-b+c= 0 ]KVL/Jan19
Exercise 21.[Row and column spaces of a matrix]Consider the matrixA=1-1013-3.(a) Isb=123in the column space ofA?(b) Isw= [4 5] in the row space ofA?(c) Describerow(A) andcol(A).Exercise 22.[Linearly Indepedence]Determine whether the following are linearly independent or dependent.Justify youranswers.111,011and10.(b) The matrices1221,-111-1and2211.(c) The polynomialsp(x) = 1 +x, q(x) = 1-xandh(x) = 1-x2.Exercise 23.[Span]Answer the following and justify your answers.15in the space spanned by1and240?(b) Is 2-3x+x2in the space spanned by 1 +xand 1 +x2?
[Linear Combination, Linearly Independence]Ifw1,w2,w3are independent vectors.Show thatv1=w2+w3,v2=w1+w3andv3=w1+w2are independent. (Hint: writec1v1+c2v2+c3v3= 0 in terms ofw1,w2,w3,and solve forc1, c2, c3.)Exercise 25.[Rank and nullity]IfAis a 3×5 matrix, explain why the columns ofAmust be linearly dependent. Whatare the possible values ofnullity(A)?[nullity(A) = 2,3 or 4 ]KVL/Jan19
EE2007/IM2007: Tutorial 4A1Additional Questions for Tutorial 4Linear Combinations, Independence, Span, Bases asAx=bAbstractThe aim of this tutorial is to help students connect the mechanism of solvingAx=bproblems with concepts discussed from slides 74 onwards, i.e., concepts suchas Linear Combinations, Linear Indepdence, Span, Column-, Row-, and Null-spacesof a matrix, etc.