F10HW3 - F2010 Homework 3 Section 1.5 14 a and b Let us do...

This preview shows pages 1–2. Sign up to view the full content.

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-

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 6

F10HW3 - F2010 Homework 3 Section 1.5 14 a and b Let us do...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online