- #1
aaaa202
- 1,169
- 2
How come the rank of a matrix is equal to the amount of pivot points in the reduced row echelon form? My book denotes this a trivial point, but unfortunately I don't see it :(
aaaa202 said:I don't what it is called in english but it's the dimension of the space that the linear function maps a vector onto.
aaaa202 said:Okay yes, I should have been able to figure that out myself. But then suppose you have row reduced matrix like the one on the attached picture. As a basis for the range you choose the vectors equal to columns with pivot points -i.e. column 1,2,3. However - wouldn't it be just as good to choose 1,2 and 4? Since that'd also make a 3 pivot points.
And lastly: Would it then also work to choose any other combination of 3 vectors out of the 4?
The rank of a linear mapping is defined as the dimension of the vector space spanned by the images of the linear mapping. In other words, it is the number of linearly independent vectors in the range of the mapping.
The rank and nullity of a linear mapping are complementary. The nullity is the dimension of the null space of the mapping, which is the set of all vectors that are mapped to the zero vector. The rank-nullity theorem states that the sum of the rank and nullity is equal to the dimension of the domain of the mapping.
The rank of a linear mapping provides important information about the behavior of the mapping. It can determine whether the mapping is injective (one-to-one), surjective (onto), or bijective (both one-to-one and onto). The rank also plays a role in solving systems of linear equations and in computing determinants.
The rank of a linear mapping is not affected by elementary row operations, which include multiplying a row by a nonzero constant, interchanging two rows, and adding a multiple of one row to another. These operations only change the representation of the mapping, but the rank remains the same.
Yes, the rank of a linear mapping can be greater than the dimensions of its domain and range. This occurs when the mapping is not a square matrix, meaning the number of rows and columns are not equal. In this case, the rank is limited by the smaller dimension, but it can still be greater than the dimensions of both the domain and range.