Understanding Injectivity and Surjectivity of Transformed Vectors

[InbredDummy]

for #2

i have the matrix:
1 1
1 0
1 0
1 0
0 0
0 0
0 0

my reasoning is that when we extended the basis of U to the basis of V, we added 3 vectors (v1,v2,v3) and in order to have T(u)=T'(u), we need to send (v1,v2,v3) to zero.

however the injective part I'm not sure about. i don't know if the null space is {0}. and the surjectivity I'm not too sure about either. and in general, I'm not sure what conditions on U,V,W,T would garner T' to be injective or surjective.

for #5:
i'm not surewhat the e* elements are exactly. so I'm really lost on this one.

 

[mighty2000]

Thank you for sharing your thoughts and questions about the matrix and its properties.

Firstly, I would like to clarify that the matrix you have provided is not enough information to fully understand the properties of T. We would need to know the specific definitions of U, V, W, and T in order to determine the injectivity and surjectivity of T'.

However, based on the information provided, I can offer some general insights. In order for T' to be injective, the null space of T' must be {0}. This means that the only vector that is mapped to 0 by T' is the zero vector. In order for this to happen, there must be no non-zero vectors in the domain of T' that are mapped to 0 by T. This would require the basis of U to be linearly independent, so that the vectors v1, v2, and v3 cannot be written as linear combinations of each other.

For surjectivity, T' would need to map every element in the range of T to an element in the range of T'. This would require the basis of W to span the entire range of T, so that every element in the range of T can be written as a linear combination of the basis vectors of W.

As for #5, the e* elements are the basis vectors of the dual space of V, denoted as V*. The dual space is the set of all linear functionals, or linear maps from V to the field of scalars. In order for T' to be an isomorphism, it must be an injective and surjective linear map. This means that for every non-zero element in V*, there must be a unique element in V that maps to it. This would require the basis of V to be linearly independent and span the entire space of V*. I hope this helps clarify your understanding. Thank you for your interest in this topic.