Linear Algebra Proof: Rank and Zero Matrix

In summary, the conversation discusses the relationship between two matrices, A and B, where A is a c x d matrix and B is a d x k matrix. It is given that the rank of A is d and that the product of A and B equals 0. The goal is to show that B must also equal 0. The solution involves representing B as a column of d elements, multiplying A and B together, and analyzing the resulting column. This proof may seem counterintuitive, but it highlights the fact that matrices are not numbers and therefore cannot be treated in the same way. Ultimately, it is concluded that because A is not the zero matrix, B must equal 0 in order for AB to equal 0.
  • #1
jumbogala
423
4

Homework Statement


A is an c x d matrix. B is a d x k matrix.

If rank(A) = d and AB = 0, show that B = 0.

Homework Equations


The Attempt at a Solution


My textbook has a solution but I don't understand it:

The rank of A is d, therefore A is not the zero matrix. (I asked my prof why d can't be equal to zero, he said it just couldn't...?)

If you left multiply A by some elementary matrix to bring it to row echelon form, you get a matrix that looks like:
[ 1 * * * ... *
0 1 * * ... *
0 0 1 * ... *
0 0 0 0 ... 0] (NOTE: * are arbitrary numbers)

And we will write B as a column (1 x k), consisting of [B1, ... , Bd]T

Multiply A and B together, and you get a column that looks like [R1, R2, ... 0, 0, 0]T

For AB = 0, then Ri = 0. Then since A is not zero, B is 0.

This proof seems to make no sense. Why are we writing B as 1 x k? It says in the question B is d x k! Also if A is not zero then why can't you say right off the bat that AB = 0 implies B =0?
 
Physics news on Phys.org
  • #2
jumbogala said:
Also if A is not zero then why can't you say right off the bat that AB = 0 implies B =0?

because these are matrices not numbers. for example
Code:
A= [0 1
   0 0]B=[1 0 
   0 0]
AB=0 yet neither A or B are 0.as to why they say 'write B as 1xk', maybe they mean write Bv (i.e. B times an arbitrary vector) as a 1xk?
 
Last edited:
  • #3
But when A is in row echelon form and you multiply it by some B, the because the solutions are zero the entries of B must be zero??
 

What is the definition of rank in linear algebra?

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. In other words, it is the dimension of the vector space spanned by the rows or columns of the matrix.

How is the rank of a matrix determined?

The rank of a matrix can be determined by performing row or column operations on the matrix to bring it into reduced row-echelon form. The number of non-zero rows or columns in this form is the rank of the matrix.

What is the significance of rank in linear algebra?

The rank of a matrix is important because it provides information about the linear independence of the rows or columns of the matrix. It also determines the dimension of the vector space spanned by the matrix, which has applications in areas such as machine learning and data analysis.

What is the relationship between rank and invertibility of a matrix?

A square matrix is invertible if and only if its rank is equal to the number of rows (or columns) in the matrix. This means that a matrix with full rank is invertible, while a matrix with less than full rank is not invertible.

Can the rank of a matrix change when performing row or column operations?

No, the rank of a matrix remains the same regardless of the row or column operations performed on it. This is because these operations do not change the linear independence of the rows or columns of the matrix, which is what determines the rank.

Similar threads

  • Calculus and Beyond Homework Help
Replies
24
Views
672
  • Calculus and Beyond Homework Help
Replies
1
Views
171
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
462
  • Calculus and Beyond Homework Help
Replies
10
Views
951
  • Calculus and Beyond Homework Help
Replies
8
Views
745
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
25
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
558
  • Calculus and Beyond Homework Help
Replies
7
Views
339
Back
Top