Practice question · True or false
If T is linear and T(u) = T(v) for two different vectors u and v, then T(u - v) is the zero vector.
Hints
- Apply linearity to the difference u - v.
- T(u - v) = T(u) - T(v), and the two terms are equal.
Show the answer
True
Why
True: T(u - v) = T(u) - T(v) = 0, and u - v is nonzero because u and v differ. So a linear map that collides on two inputs necessarily sends some nonzero vector to zero, this is exactly the kernel, and it is why injectivity will reduce to a kernel condition in the next lesson.
Practise Linear Transformations
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