Question

Given \[B=\left[\begin{matrix}2 & 1 \\ 3 & -1\end{matrix}\right]\]Find the matrix of the linear transformation \[g:M _{2}(\mathbb{R})\rightarrow M _{2}(\mathbb{R})\] given by \[g(A)=AB\] with respect to the basis \[E _{11}=\left[\begin{matrix}1 & 0 \\ 0 & 0\end{matrix}\right], E _{12}=\left[\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right],E _{21}=\left[\begin{matrix}0 & 0 \\ 1 & 0\end{matrix}\right],E _{22}=\left[\begin{matrix}0 & 0 \\ 0 & 1\end{matrix}\right]\]

Answer 1

When you have a transformation, and you want the matrix representing that transformation, you need to calculate the transformation of the standard basis "vectors". So what you need to calculate is:
\[g(E_{11}), g(E_{12}), g(E_{21}), g(E_{22})\]

Answer 2

oh okay. does the \[E _{11},E _{12}E _{21},E _{22}\] have any significance? does the 11, 12, 21 and 22 represent anything?

Answer 3

thats just letting you know where the "1" inside the matrix is.
"11" means "first row, first column"
"21" means "second row, first column"
"ij" means "ith row, jth column"

Answer 4

oh all right. and does B represent the transformation?

Answer 5

joe can u do this http://openstudy.com/groups/mathematics#/groups/mathematics/updates/4e4f59c60b8b958804a68b7e

i am kind of stuck

Answer 6

The B is pretty much the effect of the transformation. We give the transformation an input, say the matrix A, and what the transformation does is calculates AB.

Answer 7

oh wait. so you mean that we're transforming all the E matrices into a new matrix by using B? and i have found the g(E) already. not sure why they wrote the matrix in the form of columns :/

Answer 8

here is what i got for the g(E)s. i worked the first one out in full, then just put the answers for the other 3.

Answer 9

yeah i got that. but the answer they gave me was to write everything in 1 big matrix where E11 was written in the first column E12 in the 2nd column and so on.

Answer 10

as columns? thats a little odd. do you mind posting the answer so i can see it?

Answer 11

yeah sure. hold on a sec.

Answer 12

alright, i see where they are going now. so after you computed g(E_11) we got:

2 1
0 0

Thats the first column of the answer. the second third and fourth columns are g(E_12), g(E_21), and g(E_22).

So they basically created a 4x4 matrix that does the work of the transformation.

lets say you had a 2x2 matrix A equal to:

a b
c d

Then you could either compute the transformation by doing g(A) = AB, or by making a 4x1 vector (a, b, c, d) and multiplying it with that 4x4 matrix

Answer 13

oh. so it is just a different way of expressing it rather than having 4 smaller matrices?

Answer 14

right. sometimes people dont like dealing with a "vector" space of 2x2 matrices. so instead they can work with the vector space R^4, which is a little more natural to work in.

Answer 15

Given \[ f(e _{1})=(1,-1,0), f(e _{2})=(0,1,-1),f(e _{3})=(1,0,-1)\]. What is the rank and nullity?

Answer 16

If f is a column vector then it yields the following 3X3 matrix in R^3
1 0 1
-1 1 0
0 -1 -1
Add row 2 with row 1

1 0 1
0 1 1
0 -1 -1

Add row3 with row 2

1 0 1
0 1 1
0 0 0

Leaving 2 non-zero rows
Thus rank = 2

Answer 17

im not sure about nullity?

Answer 18

ahh i found it
rank + nullity = N
since rank = 2, and N=3
nullity = 1

Answer 19

nice :) i wasnt sure about it either. i dont see that terminology that much.

Answer 20

no i did a little research
linear algebra was not my fav subject :(