Department of Mathematics, University of Toronto
MAT223H1F - Linear Algebra I
Fall 2015
Tutorial Problems 1
1.Consider the system of linear equationsx1-3x2= 2-2x1+ 6x2=-4(a)Write the augmented matrix of the system, find all solutions, and express your answer in parametricform.(b)Give two di↵erent particular solutions to the given system (your choice). Call your solutionsxandy.Show thatx+y2is a solution to the system butx+yis not.(c)Prove that ifxandyare two solutions to the given system, thenax+byis also a solution if and onlyifa+b= 1.(d)Prove that, in general, a system of linear equations cannot have exactly two solutions.2.Consider the system of linear equationsax1+bx2=cbx1+cx2=dwherea6= 0.Find conditions on the remaining constantsb, c, dsuch that the system has a uniquesolution; infinitely many solutions; or no solutions.3.A homogeneous system of linear equations has augmented matrix24abccabbca35.Show that the system has a unique solution if and only ifa+b+c6= 0, anda, b, care not all equal.4 (a)LetAbe anm⇥nmatrix, prove that 0rank(A)the minimum betweenmandn. In symbols,rank(A)min{m, n}.4 (b)Use part (a) and Theorem 2 on page 15 of the textbook to prove that if a consistent system of linearequations has more unknowns than equations, then the system has infinitely many solutions.4 (c)Find an example to show that if we omit the word ’consistent’ from the statement in part (b), thestatement is no longer true.
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