Is there a connection between row reduction and linear maps?

However, the specific linear maps given by the two matrices may not be the same, as they can have different domains or ranges. In summary, the row-reduced form of a matrix and the original matrix itself may have the same row space, but the specific linear maps given by them may not be equal.
  • #1
WWGD
Science Advisor
Gold Member
7,003
10,423
Hi, everyone:

I was wondering about the relation of a matrix , seen as a linear map,
and its row-reduced form, seen the same way.

More specifically: take a matrix M , seen as a linear map L. What
is the relation between this map L given by M, and the linear map L'
given by M', the row-reduced form of M. Are these maps L,L' equal?
Are L,L' related in some other way?.

Thanks.
 
Physics news on Phys.org
  • #3
Indeed, any invertible matrix can be row reduced to the identity matrix, precisely because they both have the entire space as image (row space).
 

What is row reduction?

Row reduction is a process used in linear algebra to transform a matrix into a simpler, more organized form. It involves performing a series of elementary row operations, such as multiplying a row by a constant or adding one row to another, to eliminate entries and create a row of zeros.

Why is row reduction important?

Row reduction is important because it allows us to solve systems of linear equations and determine important properties of a matrix, such as its rank and inverse. It also helps to simplify computations and make it easier to analyze and interpret data represented by a matrix.

What is the relationship between row reduction and linear maps?

Row reduction is closely related to linear maps, also known as linear transformations. A linear map is a function that takes a vector as input and produces a vector as output, while preserving certain properties such as linearity and proportionality. Row reduction can be used to determine the matrix representation of a linear map and to analyze its properties.

What are elementary row operations?

Elementary row operations are a set of operations that can be performed on a matrix to achieve row reduction. These operations include multiplying a row by a non-zero constant, interchanging two rows, and adding a multiple of one row to another row. These operations do not change the solutions to a system of linear equations and can be used to simplify and manipulate matrices.

Can row reduction be used to solve any system of linear equations?

Row reduction can be used to solve many, but not all, systems of linear equations. If the system is consistent (has at least one solution), then row reduction will lead to a unique solution. However, if the system is inconsistent (has no solutions) or has infinitely many solutions, row reduction will not be able to provide a unique solution. In these cases, other techniques such as Gaussian elimination may need to be used.

Similar threads

I Vector subspace and basis vectors in the context of data science
    • Linear and Abstract Algebra
Replies
6
Views
847
I Explanation of a Line of a proof in Axler Linear Algebra Done Right 3r
    • Linear and Abstract Algebra
Replies
4
Views
1K
I A = norm-preserving linear map (+other conditions) => A = lin isometry
    • Linear and Abstract Algebra
Replies
4
Views
856
B Is it invalid to redefine the sgn function in this way in a proof?
    • Linear and Abstract Algebra
Replies
15
Views
4K
I Dimension and solution to matrix-vector product
    • Linear and Abstract Algebra
Replies
4
Views
852
Given specific v, dimension of subspace of L(V, W) where Tv=0?
    • Calculus and Beyond Homework Help
Replies
14
Views
582
Null space of dual map of T = annihilator of range T
    • Calculus and Beyond Homework Help
Replies
0
Views
441
I Finding the Kernel of a Matrix Map
    • Linear and Abstract Algebra
Replies
11
Views
2K
I If L is diagonalizable, algebraic & geometric multiplicities are equal
    • Linear and Abstract Algebra
Replies
4
Views
2K
MHB Mapping linear spaces to nonlinear ones.
    • Linear and Abstract Algebra
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top