I Existence and Uniqueness of Inverses

jolly_math
Messages
51
Reaction score
5
Existence: Ax = b has at least 1 solution x for every b if and only if the columns span Rm. I don't understand why then A has a right inverse C such that AC = I, and why this is only possible if m≤n.

Uniqueness: Ax = b has at most 1 solution x for every b if and only if the columns are linearly independent. I don't understand why then A has a n x m left inverse B such that BA = I, and why this is only possible if m≥n.

Could anyone explain the logic behind this? Thank you.
 
Physics news on Phys.org
Let's start with uniqueness. Suppose A has two columns that are linearly dependent. Can you find x that's not equal to 0 with Ax=0? Hint: it only has two nonzero elements
 
Office_Shredder said:
Let's start with uniqueness. Suppose A has two columns that are linearly dependent. Can you find x that's not equal to 0 with Ax=0? Hint: it only has two nonzero elements
No. A would be a column vector, and only x=0 would work. Why does this lead to the left inverse B such that BA = I, and why it is only possible if m≥n?

Thank you.
 
jolly_math said:
No. A would be a column vector

It's not. You might have misread my question, give it another look :)
 
I'm not sure what the answer is, could you explain the reasoning? Thank you.
 
Can you write out a 2x2 matrix which has two columns that are linearly dependent?
 

Thread '##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?'

I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Uniqueness of reduced row echelon form (proof verification)
Replies
4
Views
1K
I Can one find a matrix that's 'unique' to a collection of eigenvectors?
Replies
33
Views
651
I Proving an inverse of a groupoid is unique
Replies
4
Views
2K
I Dimension and solution to matrix-vector product
Replies
4
Views
3K
I Questions about non existence of GCDs vs (coimages, cokernels)
Replies
12
Views
111
I A regular matrix <=> mA isomorphism
Replies
11
Views
2K
I What is the size of the quotient group L/pZ^m?
Replies
1
Views
2K
MHB Solving Linear Equations: $Ax=b$ and Rank of A
Replies
9
Views
2K
I Dfns group action & group, written in terms of the other
Replies
26
Views
368
I How can I convince myself that I can find the inverse of this matrix?
Replies
34
Views
3K

Hot Threads

Recent Insights

Back
Top