F10HW3 - F2010, Homework 3. Section 1.5. 14. a) and b) Let...

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F2010, Homework 3. Section 1.5. 14. a) and b) Let us do it directly for an m × n matrix A and the unit vectors e j . Recall that e j is a vector in R n all the coordinates of which are 0, but for the j th which is equal to 1. We use the help of the formula c i,j = n k =1 a i,k b k,j to compute the i th coordinate of A e j and we get ( A e j ) i = n k =1 a i,k b k,j = a i,j . This precisely means that A e j = a j , j th column in the matrix A and this holds for j = 1 ,...,n . c) If A x = 0 for all vectors x R n , it is true in particular for the e i ,i = 1 ,...,n . From question b) we have that all the A e i = a i are 0 : all the columns of A are 0 which precisely means that A = O . 17. Using 14 b) we have: AI n = A [ e 1 ... e n ] = [ A e 1 ...A e n ] = [ a 1 ... a n ] = A For I m A we go back to the formulas of page 59 and we remember that in I m , the diagonal elements I i,i are equal to 1, the others I i,k with i ̸ = k are 0. The formula gives for the product c i,j = 1 × a i,j + 0 × k ̸ = i a k,j . Hence the result. 21) It is important to understand the question: AB = BA only if A and B are square matrices of the same size, means that ( AB = BA ) implies (“ A and B are square matrices of the same size”) If you are not convinced, let us say the following. Formally, if A and B are not square matrices of the same size, the equality AB = BA cannot hold that is: A and B are NOT square matrices of the same size” implies “NOT ( AB = BA )” or by considering the contrapositive: NOT(NOT AB = BA )implies NOT( “ A and B are NOT square matrices of the same size”) Nota: The statement we are asked to prove certainly does not mean that the equality holds if A and B are square matrices of the same size which would be a false statement easy to disprove by giving a counter example. Let A be an m × n matrix, and B be a p × q matrix. For the equality AB = BA to hold, both products AB and BA must exist and be of the same size. We must have n = p for the existence of the product AB which will be of size m × q , and q = m for the existence of the product BA which will be of size p × n . If both products exist, n = p and q = m , they are square and of size m × m (actually m × q but q = m ), and n × n (actually p × n but n = p ) respectively. If these products are equal they have the same size, so that m = n = p = q . The matrices A and B are both square and they have the same size. Although very simple, this exercise is already subtle enough, read carefully the so-
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F10HW3 - F2010, Homework 3. Section 1.5. 14. a) and b) Let...

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