Are the following properties true? Linear Algebra

In summary, the conversation discusses whether the following statements are true or false: a) rank(A) = rank(At) and b) dim(Nul(A)) = dim(Nul(At)). It is noted that the transpose of a matrix swaps its rows and columns, making the two statements interchangeable. However, it is clarified that both statements are false. The conversation also mentions the rank-nullity theorem and the concept of row rank equaling column rank, which is considered well-known. One of the participants suggests saving the question for when the concept of RowSpace is covered next term, while another explains that the rank of a matrix is simply the number of linearly independent columns.
  • #1
flyingpig
2,579
1

Homework Statement



Let A be a matrix

Then the following is true or false. (No need to explain why)

a) rank(A) = rank(At)
b) dim(Nul(A)) = dim(Nul(At)



The Attempt at a Solution



They are both true false right? The transpose swaps their row and columns which makes them interchanged

rank(A) = dim(Nul(At)

rank(At) = dim(Nul(A))
 
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  • #2
flyingpig said:
They are both true false right?

Greetings! I'm somewhat confused. Consider a 2 by 3 matrix; is rank(A) = rank(AT)?
 
  • #3
I meant to say "They are both false" right?
 
  • #4
Undoubtedly0 said:
Greetings! I'm somewhat confused. Consider a 2 by 3 matrix; is rank(A) = rank(AT)?

Row rank does equal column rank, doesn't it? Isn't this fairly well known?
 
  • #5
I have not learned RowSpace yet sorry, we cover that next term
 
  • #6
flyingpig said:
I have not learned RowSpace yet sorry, we cover that next term

Then you maybe haven't done the rank-nullity theorem either. Maybe you should save this question for next term?
 
  • #7
Dick said:
Row rank does equal column rank, doesn't it? Isn't this fairly well known?

Right right. I guess the point is that the rank of a matrix is simply the number of linearly independent columns (any 3x2 and any 2x3 matrix can both have at most 2 linearly independent columns). Does this help at all flyingpig?
 

What is linear algebra?

Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between those spaces. It also involves the study of systems of linear equations and their solutions.

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