Question 1205507: If TA is multiplication by a matrix A with three columns, then the kernel of TA is one of four possible geometric objects. What are they? Explain how you reached your conclusion
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Understanding the Kernel
The kernel of a linear transformation, in this case, the linear transformation induced by matrix A, is the set of all vectors that, when multiplied by A, result in the zero vector. Geometrically, it represents the subspace of the domain that is mapped to the origin of the codomain.
Possible Geometric Interpretations
Given that A is a 3x3 matrix, the domain of TA is R³. The kernel of TA can be:
A Point:
This occurs when the kernel only contains the zero vector. Geometrically, this is a single point in R³.
Algebraically, the matrix A must have full rank, meaning its columns are linearly independent.
A Line:
This occurs when the kernel is one-dimensional. It's a line passing through the origin in R³.
Algebraically, the matrix A must have rank 2, meaning two of its columns are linearly independent.
A Plane:
This occurs when the kernel is two-dimensional. It's a plane passing through the origin in R³.
Algebraically, the matrix A must have rank 1, meaning only one of its columns is linearly independent.
All of R³:
This occurs when the kernel is three-dimensional. Every vector in R³ is mapped to the zero vector.
Algebraically, the matrix A must be the zero matrix.
Conclusion
The kernel of a linear transformation induced by a 3x3 matrix can be interpreted geometrically as one of the following: a point, a line, a plane, or the entire space R³. The specific geometric interpretation depends on the rank of the matrix A, which determines the dimension of the kernel.
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