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MMLU / Suppose V is a real vector space of finite dimension n. Call the set of matrices…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Suppose V is a real vector space of finite dimension n. Call the set of matrices from V into itself M(V). Let T be in M(V). Consider the two subspaces U = {X ∈ M(V) : TX = XT} and W = {TX − XT : X ∈ M(V)}. Which of the following must be TRUE?
I. If V has a basis containing only eigenvectors of T then U = M(V).
II. dim (U) + dim (W) = n^2
III. dim (U) < n

choices

  1. I only
  2. II only
  3. III only
  4. I and II only

subject

college_mathematics

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Source and history

task_id
f71719d6-db7f-57a3-b9fc-7b760617355e
task_key
all--test--f71719d6-db7f-57a3-b9fc-7b760617355e
task_revision_id
1
config / split
all / test

Official source

initial import