Are the following properties true? Linear Algebra

[flyingpig]

Homework Statement

Let A be a matrix

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

a) rank(A) = rank(A^t)
b) dim(Nul(A)) = dim(Nul(A^t)

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(A^t)

rank(A^t) = dim(Nul(A))

 

[Undoubtedly0]

They are both true false right?

Greetings! I'm somewhat confused. Consider a 2 by 3 matrix; is rank(A) = rank(A^T)?

 

[flyingpig]

I meant to say "They are both false" right?

 

[Dick]

Greetings! I'm somewhat confused. Consider a 2 by 3 matrix; is rank(A) = rank(A^T)?

Row rank does equal column rank, doesn't it? Isn't this fairly well known?

 

[flyingpig]

I have not learned RowSpace yet sorry, we cover that next term

 

[Dick]

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?

 

[Undoubtedly0]

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?