Exercise · ZW-RANK-NULLITY-THEOREM-006 · ★★☆ Intermediate
Let \(T(x,y,z)=(x-y,\;y-z,\;x-z)\). What is the dimension of its image?
Choose one option.
Show answer
- A. \(2\)
Explanation
The third component is the sum of the first two, so the image does not fill \(\mathbb{R}^3\). The kernel is the line \(x=y=z\) (dimension \(1\)) and \(3-1=2\).
Review the concept: Rank–nullity theorem
For a linear transformation \(T\) with finite-dimensional domain \(V\), the dimension of the kernel plus that of the image equals the dimension of \(V\).
\[\dim V=\dim\ker T+\dim\operatorname{Im}T\]
More practice problems: rank–nullity theorem
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