Thinking about matrices/polynomials

[eep]

Hi,
A homework problem I ran across awhile ago asked me to determine if a set of of 2x2 Matrices were a basis for the set of aa 2x2 matrices.

Am I going to run into any pitfalls by thinking about such 2x2 matrices as vectors of 4 components? Basically what I did was turn each 2x2 matrix into a 4x1 vector. Each row represented an entry in matrix A (row 1 was A11, row 2 was A12, row 3 was A21, row 4 was A22).

Basically, I have no problems in dealing with vectors but when I run across problems where I'm given either polynomials or matrices with columns I'm unsure as to how I can approach them. For polynomials I figure I can just treat each power of x as a separate component of a vector. Any insight would be appreciate and sorry if this post is jibberish, I'm a little tired. Thanks!

 

[HallsofIvy]

As long as you are dealing with matrices as a vector space you are not using matrix product so, yes, you can just think of 2x2 matrices as a 4 dimensional vector.

 

[Hurkyl]

Am I going to run into any pitfalls by thinking about such 2x2 matrices as vectors of 4 components? Basically what I did was turn each 2x2 matrix into a 4x1 vector. Each row represented an entry in matrix A (row 1 was A11, row 2 was A12, row 3 was A21, row 4 was A22).

It sounds like you're just using the coordinates defined by the ordered basis:

$$\left( \left[\begin{array}{ll}1 & 0 \\ 0 & 0 \end{array}\right] , \left[\begin{array}{ll}0 & 1 \\ 0 & 0 \end{array}\right] , \left[\begin{array}{ll}0 & 0 \\ 1 & 0 \end{array}\right] , \left[\begin{array}{ll}0 & 0 \\ 0 & 1 \end{array}\right] \right)$$

and using coordinates is fine, though not always the most efficient method of working with vectors.


(Yes, $$\left[\begin{array}{ll}1 & 0 \\ 0 & 0 \end{array}\right]$$ is a vector, and so is $x^3 - 4x + 17$. You sound like you might be confusing yourself by using "vector" as a synonym for "n-tuple")