471-QuestionBank

• Is there a relationship between the composition of functions and the ability to be able to perform matrix multiplication among similar vector spaces in any order?
• The commutativity of matrix multiplication

$$F^n\to F^n\to F^n$$ $$[T_a{]}_B^B\circ [T_b{]}_B^B$$

• What do the dotted lines in the commutative diagrams mean?

• What does it mean when a diagram is commutative?

• How do we define Translations in terms of Matrix Multiplication?

• Is there a particular reason we examine the transformations where $W\ne V$. Are the $End(V)$ just not that interesting to examine?

• When we define a transformation $T:V\to W$, we get a natural isomorphism through through theorem 1. where its an isomorphism that maps elements in to $ImT$. What part of this theorem does this, what is this specific isomorphism? any more information can be extrapolated from it?

  • Suppose we have two equivalent system, are the isomorphism exactly the same? They do map each other to the exact same line!, but may have a different kernel!
  • Is there any relation to the solution space and the ImT? If they are the same, does that mean both systems are equivalent?
  • Where does the “projection aspect” come from in the example?

• In the equivalent systems, we can observe that the two systems are the same by ERO operations, however, their images are different, is there a way to observe that they are the same? Just by looking at the information we are given?

  • Obviously if they were homogenous, we can see the kernels are exactly the same! but if they were not homogenous, how can we see if they are equivalent? Do they have the same preimages? Does this relate the the isomorphism between ImT and preimage of T?

• Diagonal matrices are always invertible

  • Given a matrix $\left(\begin{array}{ccc}a& & \\ & \ddots & \\ & & m\end{array}\right)$, there is an inverse for it!
  • How can you compute the number of matrices that perform the same operation: For example, the following matrices are equivalent : $\left[\begin{array}{ccc}a& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\sim \left[\begin{array}{ccc}0& 0& 0\\ a& 0& 0\\ 0& 0& 0\end{array}\right]\sim \left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ a& 0& 0\end{array}\right]$ As the all scale the “first basis vector” by a

• Why does Lecture 25 The Dual Vector Space or Matrix Transposition proof of dual space look a lot like matrix multiplication?

• Find a way to easily reindex summations

• Why is it called a symmetrizer?