What are the kernel and range of the linear map T(x₁,x₂,x₃)=(x₁−x₂,2x₃)?
Conditions
- The linear map is defined as T(x₁,x₂,x₃)=(x₁−x₂,2x₃).
- The domain is three-dimensional and the codomain is two-dimensional.
Reasoning, step by step
- Identify the kernel by finding vectors where .
- Solve x₁−x₂=0 and 2x₃=0 to find the basis (1,1,0).
- Identify the range by observing the output dimensions.
- Conclude the range spans the entire two-dimensional codomain.
- Determine the nullity as the dimension of the kernel (1).
- Determine the rank as the dimension of the range (2).
Example
For T(x₁,x₂,x₃)=(x₁−x₂,2x₃), the kernel is spanned by (1,1,0) and the range is all of the two-dimensional codomain. The nullity is one and the rank is two.
Common misconceptions
- Assuming the kernel contains more than one basis vector.
- Believing the range is a proper subset of the codomain when it is actually the entire space.
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