Kernel, Range, Basis (linear algebra)

In summary, the kernel is {(1, -1, 0)}, the range is all of R2, and the basis is the standard basis of {(1,0),(0,1)}
  • #1
Codexmac
4
0
Hey all!
I am working on this and got confused. Any help at all would be much appreciated!

Determine the kernel and range of the transformation T and find a basis for each: T(x,y,z)=(x,y,z) from R3 to R2.

I have found the kernel to be the set {(r, -r, 0)}.
Range is R2.
I"m not sure how to find the basis. Any ideas? I think the range is right but not sure.
Thanks guys!
 
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  • #2
The transformation your have written maps R^3->R^3.
 
  • #3
Sorry the transformation is T(x,y,z) = ( x + y, z) Any ideas what the basis is?
 
  • #4
What do you mean basis of a transformation? A basis is something associated with a subspace.
 
  • #5
He means a basis for the kernel and a basis for the range.

So you have found the range to be all of R^2, then you may give any basis for R^2, recall a basis is a set of vectors from the space such that they are linearly independent and span the space. There is a standard basis for R^2 you may use here...

Similarly, you have characterized all the vectors in the kernel by r*(1,-1,0), where r is any number in R, think about the definition of spanning a space: a set of vectors spans a space if any vector from the space can be expressed as a linear combination of those vectors. So now can you tell me what is the basis you have found for the kernel?
 
  • #6
Codexmac said:
Sorry the transformation is T(x,y,z) = ( x + y, z) Any ideas what the basis is?
The kernel consists of all (x,y,z) such that T(x,y,z)= (x+y, z)= (0, 0). That is, x+y= 0 or y= -x and z= 0. Any vector in the kernel is of the form (x, -x, 0)= x(1, -1, 0) and so {(1, -1, 0)} is a basis for that one-dimensional vector space.

The range consists of all numbers of the form (x+y, z). Since x and y can be any numbers, so can x+y. The range is all of R2 and so has {(1, 0), (0, 1)} as basis.

Notice that the two dimensions, 1 and 2, add to 3, the dimension of R3.
 
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  • #7
Okay that makes sense. Thanks guys! The basis is the standard basis of {(1,0),(0,1)} And the range can be any number in R2.
and yes the dim Ker(T) is 1, dim Range(T) is 2, dim Domain(T) is 3.
 

1. What is a Kernel in linear algebra?

A Kernel, also known as null space, is the set of all vectors that when multiplied by a specific matrix result in a zero vector. In other words, it is the solution space to the homogeneous equation Ax=0.

2. How is the Kernel related to linear independence?

In linear algebra, the Kernel is closely related to linear independence. If a vector is in the Kernel of a matrix, it means that it is a linear combination of other vectors in the matrix. This implies that the vector is dependent on the other vectors in the matrix and not linearly independent.

3. What is the Range of a matrix?

The Range, also known as the image, of a matrix is the set of all possible outputs that can be obtained by multiplying the matrix by any input vector. In other words, it is the span of the columns of the matrix.

4. How is the Range related to the columns of a matrix?

The Range is closely related to the columns of a matrix. The columns of a matrix span the Range, which means that any vector in the Range can be written as a linear combination of the columns of the matrix.

5. What is a Basis in linear algebra?

A Basis is a set of linearly independent vectors that span a vector space. It is the smallest set of vectors that can be used to represent all other vectors in the space. The number of vectors in a Basis is called the dimension of the vector space.

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