IN SECTION3.5 PROBLEM 37
The Question: The subspace of matrices that commute with the shift S has dimension__ .

The Solution:Since there are three free variables, the subspace of these matrices has dimension 3

Can someone explain what dose the solution mean? I am confused.

Question

IN SECTION3.5 PROBLEM 37 
The Question: The subspace of matrices that commute with the shift S has dimension__ .

The Solution:Since there are three free variables, the subspace of these matrices has dimension 3

Can someone explain what dose the solution mean? I am confused.

jerrychan · Answer

Problem and Solution

phi · Answer

First, their solution for A has a typo. The bottom right entry should be just "a" not ai

A is characterized by 3 numbers, namely a , along the main diagonal, b and c
If we denote a point in 3-dim space by the coords (a,b,c), every point corresponds to a particular matrix A (of this specific form).  In other words, it takes 3 independent numbers to specify a specific A matrix, and those three numbers correspond to 3 dimensions.

jerrychan · Answer

THANKS YOU,phi. 
Two more question: So the three variables are a,b and c,right?
Does the expression, the subspace of the matrices, refer to the matrix space (the set contains all A)?
IF it is, can the three matrix below be the basis of the matrix space?
|dw:1451694518893:dw|