# Linear Algebra Matrix Proof problem

• JKCB
In summary, Office_Shredder is trying to solve a homework problem involving inverting an n x m matrix with rank m. He is stuck and needs help from an expert.
JKCB

## Homework Statement

Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB=Im

## The Attempt at a Solution

I'm assuming I would need to start with the def. That there exists P an mxm invertible matrix and Q an nxn invertible matrix s.t. A=P(Im 0)Q

then P(Im 0)Q B = Im

now I might multiply left hand side by P inverse and right hand side by Q inverse.

I'm stuck am I going in the right direction?

I'm a bit confused by the notation here. What is I am 0? I'm assuming I am is the mxm identity matrix

Yes I am is the mxm identity matrix and 0 is the mxn zero matrix.

I'm thinking of going another direction.

What if I start with letting B be any nxm matrix with rank n. Then AB would be an mxm matrix with rank m, then by the Thm (in my book) 2.18 corollary 2 an nxn matrix is invertible iff its rank is n. AB is invertible. Let M be the inverse then (AB)M=Im which means
A(BM)=Im therefore BM would be the nxm matrix we are looking for.

What do you think? Would that do it, any holes?

JKCB said:
Yes I am is the mxm identity matrix and 0 is the mxn zero matrix.
As Office_Shredder asked, what does I am 0 mean? Inquiring minds want to know.

Im (I subscript m) is the mxm identity matrix and 0 is the m x (n-m) zero matrix.

Then Im 0 is the product of Im and the m x (n - m) zero matrix, which is 0. Were you thinking that Im times a zero matrix is something other than the same zero matrix?

No. How about this? A= (Im 0)P where P is an nxm invertible matrix. Then replace A with (Im 0)P(B) = I am then (Im 0) P P^-1(Im)=Im
(0 ) (that is a column block matrix with I am being the Identity matrix and 0 being a zero matrix (n-m) x m) then that would make
B = p^-1 (Im)
(0 ) That is a partitioned matrix I am is mxm and the zero is (n-m) x m
Will that work?

(that is a column block matrix with I am being the Identity matrix and 0 being a zero matrix (n-m) x m

Is this what I am 0 is? It's hard to tell

I think I have figured out what you're trying to communicate, but your notation was no help. What you are writing as (Im 0) looks to me like a matrix product, and what you meant was the m x n matrix (Im|0).

## 1. What is a "Linear Algebra Matrix Proof problem"?

A Linear Algebra Matrix Proof problem is a mathematical question that involves using concepts and techniques from linear algebra and matrices to prove a statement or solve a problem. It typically requires an understanding of matrix operations, properties, and theorems.

## 2. What are the steps involved in solving a Linear Algebra Matrix Proof problem?

The steps involved in solving a Linear Algebra Matrix Proof problem include:

1. Understanding the given statement or problem
2. Identifying the relevant concepts and theorems from linear algebra
3. Using these concepts and theorems to manipulate the matrices or vectors involved
4. Applying logical reasoning to prove the given statement or solve the problem
5. Writing a clear and concise proof, showing all the steps and justifications

## 3. What are some common techniques used in Linear Algebra Matrix Proof problems?

Some common techniques used in Linear Algebra Matrix Proof problems include:

• Properties of matrix operations, such as commutativity and associativity
• Linear independence and linear dependence
• Inverse matrices
• Eigenvalues and eigenvectors
• Rank and nullity of a matrix

## 4. How can I improve my skills in solving Linear Algebra Matrix Proof problems?

To improve your skills in solving Linear Algebra Matrix Proof problems, you can:

• Practice solving various types of problems from textbooks or online resources
• Review and understand the concepts and theorems involved
• Work with a study group or seek help from a tutor or professor
• Try to come up with your own examples and proofs to solidify your understanding
• Read and analyze sample proofs to learn different approaches and strategies

## 5. What are some real-world applications of Linear Algebra Matrix Proof problems?

Linear Algebra Matrix Proof problems have various real-world applications, including:

• Computer graphics and image processing
• Machine learning and data analysis
• Engineering and physics, such as in solving systems of linear equations
• Finance and economics, for modeling and analyzing data
• Robotics and control systems
• Cryptography, for designing and breaking codes

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