Rank(A) + nullity(A) = no. of cols of A (WHY?)

  • Thread starter nyxynyx
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In summary: Gaussian elimination will give you a system of equations that is exactly the rank-nullity theorem equation.
  • #1
nyxynyx
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Hello! I am confused over why rank(A) + nullity(A) = n = no. of columns of A, not no. of rows or something else.

My lecturer showed me something like a mxn matrix postmultiplied with a x-vector to get R^n, thus n = no. of cols. Makes sense when he was explaining but when i stepped out i realized that i didnt get it. Any help pls? Thanks!
 
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  • #2
You can probably find a proof somewhere online.

It's pretty quite intuitive after you get further into Linear Algebra and have become more comfortable with Nullspaces and such. I'll find a link with a proof.
 
  • #3
I can't one what doesn't use linear transformations!
 
  • #4
My module hasn't reached transformations yet :(. Is there a explanation why its equal to no. of columns without talking about transformation?
 
  • #5
In my humble opinion, the rank-nullity theorem is not something you really want to "explain" -- it should be part of the foundation of your intuition for linear algebra. If you're looking for understanding, your best bet is probably to review previous exercises where you actually solved systems of equations and apply the rank-nullity theorem to describe the system.

e.g. you may have done an exercise solving Ax=b, where A is 3x3, and got a two-dimensional space of solutions. The rank-nullity theorem says that the rank of A is one -- so confirm that by computing the rank of A!
 
  • #6
Hurkyl said:
In my humble opinion, the rank-nullity theorem is not something you really want to "explain" -- it should be part of the foundation of your intuition for linear algebra. If you're looking for understanding, your best bet is probably to review previous exercises where you actually solved systems of equations and apply the rank-nullity theorem to describe the system.

e.g. you may have done an exercise solving Ax=b, where A is 3x3, and got a two-dimensional space of solutions. The rank-nullity theorem says that the rank of A is one -- so confirm that by computing the rank of A!

Exactly. It comes around. If it hasn't yet, continue solving systems. :biggrin:
 
  • #7
Once you learn something about linear operators and their matrix representation, it should become formally clear.

Edit: actually, you can investigate this fact by going into Gaussian elimination.
 
Last edited:

FAQ: Rank(A) + nullity(A) = no. of cols of A (WHY?)

1. How do you define "Rank" and "nullity" in the context of matrices?

In linear algebra, the rank of a matrix refers to the maximum number of linearly independent rows or columns in the matrix. The nullity of a matrix is the dimension of the null space, which consists of all vectors that, when multiplied by the matrix, result in a zero vector.

2. Why does the sum of the rank and nullity of a matrix always equal the number of columns?

This is due to the Rank-Nullity Theorem, which states that the sum of the rank and nullity of a matrix is equal to the number of columns in the matrix. This is because the rank of a matrix represents the number of linearly independent columns, while the nullity represents the number of linearly dependent columns. Since every column in a matrix must fall into one of these categories, their sum will always equal the total number of columns.

3. How can I calculate the rank and nullity of a matrix?

The rank can be calculated by performing row operations to reduce the matrix to echelon form and counting the number of non-zero rows. The nullity can then be calculated by subtracting the rank from the number of columns. Alternatively, the rank can also be calculated by finding the dimension of the column space, while the nullity is the dimension of the null space.

4. What does it mean if a matrix has a rank of 0?

If a matrix has a rank of 0, it means that all of its columns are linearly dependent, and therefore the matrix has no linearly independent rows or columns. This also means that the nullity of the matrix is equal to the number of columns, as there are no linearly independent columns to contribute to the rank.

5. Can the rank and nullity of a matrix be equal?

No, the rank and nullity of a matrix cannot be equal. This is because the rank represents the number of linearly independent columns, while the nullity represents the number of linearly dependent columns. Since these two values are complementary, their sum will always equal the number of columns in the matrix.

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