ECO_1010_2018_HW_A.pdf - University of Toronto, Department...

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University of Toronto, Department of Economics, ECO 1010. Ajaz Hussain1ECO 1010 Fall 2018HW for Lecture A__________________________________________________________________________________________________1)Let? = (1,2), ? = (1, −3), ? = (0,1,1).Compute the following expressions, whenever these are defined (you mayalso want to do these inWolfram Alpha):a)? + ?b)? − ?c)2? + ?d)? + ?2)Let? = (1,2,0) and ? = (0,1,1).Compute the following expressions, whenever these are defined (you may alsowant to do these inWolfram Alpha):a)2? + ??b)2? − ??3)Let? = (3, −1,0)and? = (0,1,5)a)Show that no choice of numbers?and?can make?? + ?? = (3,0,0).b)For what value(s) of?(if any) can the equation?? + ?? = (3,0, ?)be satisfied?4)We have seen thatanyset with operations of addition and multiplication by real numbers (? + ? =(?1+ ?1, … , ?𝑛+ ?𝑛)and?? = (??1, . . , ??𝑛)) defined in such a way that the following seven formulas (“laws”) holdis avector spaceand its elements are calledvectors. In this course, that ‘set’ is𝑛 −tuples of real numbers andtherefore the vector space that we almost always work with is𝑛(whose elements can now be labelled vectors).These seven formulas (“laws”) hold for any arbitrary?, ?, ?in𝑛and any arbitrary numbers (scalars)?and1.?? + ?? = (? + ?)?2.?? + ?? = ?(? + ?)3.?(??) = (??)?4.? + ? = ? + ?5.(? + ?) + ? = ? + (? + ?)6.? + ? = ?7.? + (−?) = ?and?? = ?Prove “law #2”.5)Suppose we define the addition of two vectors?, ? ∈ ℝ2as follows:.?:
University of Toronto, Department of Economics, ECO 1010. Ajaz Hussain2(?1?2) + (?1?2) = (?1+ ?2?2?2)And the scalar multiplication of a vector as:? (?1?2) = (??1??2)Recall the seven laws of a vector space:1.?? + ?? = (? + ?)?2.?? + ?? = ?(? + ?)3.?(??) = (??)?4.? + ? = ? + ?5.(? + ?) + ? = ? + (? + ?)6.? + ? = ?7.? + (−?) = ?and?? = ?a)Is law #4 satisfied?b)Is law # 5 satisfied?c)Is law #2 satisfied?d)What do you conclude about the set of vectors in2?6)Consider the set of 2×3 matrices, an element of which is:? = (?11?12?13?21?22?23) ∈ The set of 2 × 3 matricesSuppose we define addition and scalar multiplication of the elements in the set of 2×3 matrices as:? + ? = (?11?12?13?21?22?23) + (?11?12?13?21?22?23) = (?11+ ?11?12+ ?12?13+ ?13?21+ ?21?22+ ?22?23+ ?23)?? = (??11??12??13??21??22??23) where ? is a scalarNoting that(−1)? = −?

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