Geometric description of kernel

In summary, a kernel in geometric description is a set of points that are mapped to the origin by a linear transformation. It is represented as a subspace in the vector space, and its dimension is determined by the number of linearly independent vectors that make up the subspace. The kernel and the range of a transformation are complementary subspaces, and the rank-nullity theorem states that the dimension of the kernel can be found by subtracting the rank of the transformation from the dimension of the vector space.
  • #1
eyehategod
82
0
T is the projection onto the xy-coordinate plane:
T(x,y,z)=(x,y,0)

I have to give a geometric description of the kernel and range of T.


my geometric description of the kernel:
a line along the z-axis. Is this correct?
whats the geometric description of the range of T?
 
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  • #2
Why "a line along the z-axis". Why not just "the z-axis"?

Is not the range, the set of all pairs (x,y, 0) pretty obvious?
 

FAQ: Geometric description of kernel

What is a kernel in geometric description?

In geometric description, a kernel is a set of points that are mapped to the origin by a linear transformation. It is the set of all vectors that are mapped to the zero vector by the transformation.

How is a kernel represented geometrically?

A kernel is represented geometrically as a subspace in the vector space. It is a plane or line that passes through the origin and is perpendicular to the direction of the transformation.

How is the dimension of a kernel determined?

The dimension of a kernel is determined by the number of linearly independent vectors that make up the subspace. This can be found by solving the equation Ax = 0, where A is the transformation matrix and x is a vector in the kernel.

What is the relationship between the kernel and the range of a transformation?

The kernel and the range of a transformation are complementary subspaces. This means that any vector in the range of the transformation is perpendicular to any vector in the kernel, and vice versa.

How does the rank-nullity theorem relate to the kernel?

The rank-nullity theorem states that the rank of a transformation plus the dimension of its kernel equals the dimension of the vector space. This means that the dimension of the kernel can be found by subtracting the rank of the transformation from the dimension of the vector space.

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