# Proving the Isomorphism of [ ]B: L(V) to Mnxn(R) in Linear Transformations

• sweetiepi
In summary, the exercise is asking for a proof that the function [ ]B: L(V) -> Mnxn(R) is an isomorphism, where T -> [T]B is the B-matrix for T, and T is a linear transformation in the vector space of all linear transformations. This means that every matrix and linear transformation have a unique relationship, and the exercise is asking to prove this relationship. The title may have initially mentioned rank, but the focus is on proving the relationship between matrices and linear transformations.

#### sweetiepi

Prove that the function [ ]B: L(V) -> Mnxn(R) given by T -> [T]B is an isomorphism. [T]B is the B-matrix for T, where T is in the vector space of all linear transformations.

I don't quite understand this...

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Please elaborate: what don't you understand? Do you know what an isomorphism is?

What this exercise says, is that given a basis, every matrix determines a unique linear transformation, and vice versa. You probably already knew this, now you have to prove it. It really is "writing out the obvious".

PS: I don't quite see the connection with 'rank'.

Oh sorry about the confusion with the title. When I initially posted a question, it had to do with rank, but then I figured that one out so I just edited the question rather than post a new thread and must have forgotten to change the title.

I suppose my confusion was just because it's hard for me to visualize what's happening with coordinate mappings for a B-matrix for a transformation. I tend to have the most difficulty with the obvious things. But regardless, I went to talk to a GSI about it today and he actually helped a lot. I just needed help getting the problem started, but I got it now.

## What is a "Rank question"?

A rank question is a type of survey question that asks respondents to rank a list of items in order of preference or importance.

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The purpose of using rank questions is to gather data on the relative importance or preference of a list of items. This can provide insight into the opinions and priorities of the respondents.

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Rank questions should be formatted as a list of items, followed by a space for the respondent to enter their ranking or a drop-down menu to select the ranking. It is important to clearly label the items and provide clear instructions for ranking.

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One limitation of rank questions is that they can be time-consuming for respondents, especially if the list of items is long. They also do not allow for nuanced responses, as the respondent must choose a specific ranking for each item.

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Rank questions can be used in conjunction with other survey questions to gather more detailed information about the respondents' opinions and preferences. For example, a rank question can be followed by open-ended questions to gather more in-depth explanations for the ranking choices.