Max Elongation of a Vector Druing Linear Mapping

[shaofent]

Hi all,
I am studying linear algebra recently and following immersive math book. Now I am reading the section about how to determine input v gives the largest and smallest elonggation through linear mapping F(v)=Av. Here is the explanation and deduction:


However, I cannot understand these explanantions marked with red box above. How could he deduce 'null space of at least dimension 1' from previous conditions? Can somebody help me explain more about this? Thanks.

 

[HallsofIvy]

Are you clear on what the "null space" is? The null space of a linear transformation, T, from space U to space V, is the set of all vectors, w, in U, such that Tw= 0, the 0 vector in V. It is fairly easy to show that the null space is a subspace of U, perhaps of dimension 0 (the single vector 0 in U). Here the hypothesis, "Hence if \(\displaystyle w\ne 0\)", is that there exist a non-zero vector that is mapped to 0. Since there is a non-zero vector, w, in the null space it has dimension larger than 0 so "at least dimension 1".

 

[shaofent]

Are you clear on what the "null space" is? The null space of a linear transformation, T, from space U to space V, is the set of all vectors, w, in U, such that Tw= 0, the 0 vector in V. It is fairly easy to show that the null space is a subspace of U, perhaps of dimension 0 (the single vector 0 in U). Here the hypothesis, "Hence if \(\displaystyle w\ne 0\)", is that there exist a non-zero vector that is mapped to 0. Since there is a non-zero vector, w, in the null space it has dimension larger than 0 so "at least dimension 1".

Hi Hallsoflvy,

Thank you for your answer. Actually, I know that null space is the entire set of solutions to homogeneous systex Ax = 0, but it doesn't help me work out my issue. After reading your interpretation, I got something:
If the dimension of the null space is 0, then zero vector 0 must exist in the null space and it is the only vector in the null space. Otherwise, it breaks the truth that null space is a linear space. For example, if it has another vector a≠0 in the null space. To keep null space a linear space, it should have any vector ka belongs to the null space. But it is not.
So in my question with hypothesis that w≠0, the dimension of the null space could not be 0, it must be greater or equal than dimension 1.

Is my understanding correct? Thanks.