Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question

The pictures are the answer of the question and the theorem 3, I would to ask why the answer of this question can give the result that rank(ATA)=n ?

This is the question: Let A be an m×n matrix with columns c1,c2,…,cn. If rank A=n, show that {ATc1,ATc2,…,AT cn} is a basis of Rn.

Let A be m x n matrix with columns C₁, C₂, . in terms of its column: Cn ]. A = [C₁ C₂ Then A¹c, is the jth column of ATA : Cn. We can write matrix A O AT A = AT [ C₁ C2 C1 Cn] =[ATC₁ ATC₂ AT cn]. = Since we have that rank A = n, from Theorem 3 part 4 we have that ATA is nx n and it is invertible. This means than rank(ATA). = n. Applying Theorem 3 part 3 on ATA we have that columns of ATA are linearly independent in R" and applying Theorem 4 part 2 we have that they also span R". This means that {ATc₁, ATC2, ..., AT cn} is a basis of R".
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Transcribed Image Text:Let A be m x n matrix with columns C₁, C₂, . in terms of its column: Cn ]. A = [C₁ C₂ Then A¹c, is the jth column of ATA : Cn. We can write matrix A O AT A = AT [ C₁ C2 C1 Cn] =[ATC₁ ATC₂ AT cn]. = Since we have that rank A = n, from Theorem 3 part 4 we have that ATA is nx n and it is invertible. This means than rank(ATA). = n. Applying Theorem 3 part 3 on ATA we have that columns of ATA are linearly independent in R" and applying Theorem 4 part 2 we have that they also span R". This means that {ATc₁, ATC2, ..., AT cn} is a basis of R".
Theorem 5.4.3 The following are equivalent for an m × n matrix A: 1. rank A = n. 2. The rows of A span R". 3. The columns of A are linearly independent in R™. 4. The nxn matrix AT A is invertible. 5. CA In for some nxm matrix C. = = 6. If Ax=0, x in R", then x = 0.
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Transcribed Image Text:Theorem 5.4.3 The following are equivalent for an m × n matrix A: 1. rank A = n. 2. The rows of A span R". 3. The columns of A are linearly independent in R™. 4. The nxn matrix AT A is invertible. 5. CA In for some nxm matrix C. = = 6. If Ax=0, x in R", then x = 0.
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