Kernel and image of a matrix A

In summary, the kernel and image of a matrix A are the solution set and range of the linear transformation T associated with matrix A.
  • #1
Niles
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[SOLVED] Kernel and image of a matrix A

Homework Statement



If I have a matrix A, then the kernel of A is the solution to Ax=0?

The image of A is just the vectors that span the column space?

I have looked through my book and searched the WWW, but I can't find the answer to these questions anywhere. I hope you guys can help.

Thanks in advance.
 
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  • #2
Loosely interpreted, I think you got it right, but you should include the fact that A is the standard matrix associated with the linear transformation. This is a question on linear transformations, right?

Your first statement should read "The kernel of the linear transformation T, which is associated with standard matrix A, is the nullspace of A, which the the solution set of Ax=0".

Your second statement should read "The set of images of the linear transformation T, which is associated with the standard matrix A, also known as the range of T, is given by the column space of A"

I know it seems pedantic but sometimes confusion is caused when we use imprecise language in mathematics.
 
  • #3
Suppose A is a mxn matrix. View A as a linear transformation that left-multiplies a column vector in R^n to a column vector in R^m. Since A:R^n->R^m is now a linear transformation (prove it!), then by definition ker(A) is the solution space to Ax=0.

Now im(A) is a subspace of R^m (prove it!). What set generates this subspace? Well take the n standard basis vectors of R^n. Then the image of these n vectors by the linear transformation A are simply the columns of the matrix A, right? So im(A) is the span of the columns of A (since the n standard basis vectors form a basis of R^n), i.e. the column space of A.

Furthermore, suppose rankA (the dimension of im(A)) = r. Then by the dimension theorem, we have the number of linearly independent solutions to Ax=0 to be n-r, which you've probably learned in high school but now see the proof of. Also, rankA, which is the dimension of im(A)=column space of A, is now the number of linearly independent columns of A (and also the number of linearly independent rows of A).
 
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  • #4
Thanks to you both - it's so great that you guys can help.

So it doesn't make sense talking about the image and kernel of a matrix alone (and not associating it with a linear transformation)?
 
  • #5
That's right.

ker: Hom(V,W) -> V
im: Hom(V,W) -> W

where Hom(V,W) is the set of all linear transformations from a vector space V to a vector space W. So ker, and I am do not act on matrices.
 

1. What is the kernel of a matrix A?

The kernel of a matrix A is the set of all vectors that when multiplied by A result in the zero vector. In other words, it is the set of all solutions to the equation Ax=0. The kernel is also known as the null space of A and is denoted by ker(A).

2. How is the kernel related to the image of a matrix A?

The kernel and image of a matrix A are related by the fundamental theorem of linear algebra. This theorem states that the dimension of the kernel of A plus the dimension of the image of A equals the number of columns in A. In other words, the dimension of the kernel and image add up to the total number of variables in the system of equations represented by A.

3. What is the significance of the kernel and image of a matrix A?

The kernel and image of a matrix A provide important information about the linear transformation represented by A. The kernel represents the set of all inputs that result in no change or a trivial output, while the image represents the set of all possible outputs. These concepts are essential in understanding the behavior and properties of linear systems.

4. How can the kernel and image be calculated?

The kernel and image can be calculated using various methods such as Gaussian elimination, finding the nullspace using row reduction, or using the rank-nullity theorem. In some cases, the kernel and image can also be determined geometrically by visualizing the linear transformation represented by A.

5. Can the kernel and image of a matrix A be empty?

Yes, it is possible for the kernel and image of a matrix A to be empty. This occurs when the linear transformation represented by A is one-to-one, meaning that each input has a unique output. In this case, the kernel only contains the zero vector, and the image contains all possible outputs.

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