Can't Row Reduce Matrix - Finding Nullspace and Imagespace

In summary: You just have more rows than columns, so the nullspace will be bigger.In summary, the conversation discusses a matrix A and the difficulties in finding its nullspace and image space. It is suggested to use Gaussian elimination to row reduce the matrix, which can then be reduced to Jordan-form in two steps. Alternatively, the null space can be thought of as (x,y,z) satisfying certain equations. The procedure for finding the nullspace and image space is the same for nonsquare matrices as it is for square matrices.
  • #1
mr_coffee
1,629
1
Hello everyone...
I have the following matrix:
A =
-1 -4 1
7 -9 0
10 3 -3
-9 1 2

I can't row reduce this sucker! This isn't an agumented matrix i don't think, so i can't just take the square matrix and then find the inverse and mutliply it by vector b to find the values of a, b, c, d; So because i can't do this, I'm getting suck on figuring out the Nullspace and image space. What is the procedure to find the null and image space of a matrix that isn't nxn?

I'm trying to check it with maple ( a math program ) to see if there but i created the matrix, did all that good stuff now typed in Nullspace(A) mod P; what mod am i suppose to use, they didn't explain that part. But i'd rather have an explanation than a program doing it for me. Thanks.
 
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  • #2
Why can't you row reduce it? Just use Gaussian elimination. This matrix can be reduced to Jordan-form in 2 steps.
 
  • #3
Or you can just think of the null space as (x,y,z) satisfying
x- 4y+ z= 0
7x-9y= 0
10x+ 3y- 3z= 0
-9x+ y+ 2z= 0

What x, y, z satisfy all four of those?
 
  • #4
The procedure to find the image and nullspace of a nonsquare matrix should be exactly the same as for a square matrix.
 

FAQ: Can't Row Reduce Matrix - Finding Nullspace and Imagespace

What does it mean to "row reduce" a matrix?

Row reducing a matrix is a process in linear algebra where the rows of a matrix are manipulated using elementary row operations to transform the matrix into a simpler form, typically a row echelon form or reduced row echelon form. This process is used to solve systems of linear equations, find the rank and nullity of a matrix, and determine the linear independence of a set of vectors.

Why can't I row reduce a matrix?

There are a few reasons why you may not be able to row reduce a matrix. One common reason is that the matrix may not be square, meaning it does not have an equal number of rows and columns. Another reason could be that the matrix is already in reduced row echelon form and further row operations will not change it. Additionally, if the matrix is singular, meaning it has a determinant of 0, it cannot be row reduced.

What is the nullspace of a matrix?

The nullspace of a matrix, also known as the kernel, is the set of all vectors that, when multiplied by the matrix, equal the zero vector. In other words, it is the set of all solutions to the homogeneous system of linear equations represented by the matrix. The nullspace is a subspace of the vector space in which the matrix operates.

What is the imagespace of a matrix?

The imagespace of a matrix is the set of all vectors that can be obtained by multiplying the matrix by a vector from its domain. In other words, it is the span of the columns of the matrix. The imagespace is a subspace of the vector space in which the matrix operates.

How do I find the nullspace and imagespace of a matrix?

To find the nullspace and imagespace of a matrix, you can use row reduction to transform the matrix into reduced row echelon form. The pivot columns in the reduced matrix correspond to the basis vectors of the imagespace, while the non-pivot columns correspond to the basis vectors of the nullspace. The vectors in the nullspace can be found by setting the free variables to arbitrary values and solving for the pivot variables, and the vectors in the imagespace can be found by setting the pivot variables to 1 and the free variables to 0.

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