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Linear algebra, problem with linear maps and their matrices

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Homework Statement


We have the following linear map:
f:R^2-->R^3 f(x,y)=(x,3x+8y,x+y+11y)
e is the standar basis of R^2 and
a is the standar basis of R^3
Question: Is there a linear map g: R^3-->R^2 that the matrix of (fog, a,a) is the 0 matrix?
Is there a linear map g: R^3-->R^2 that the matrix of (gof, e,e) is the 0 matrix?




Homework Equations





The Attempt at a Solution



First of all the matrix of f for the standar basis is:
1 0
Α= 3 8
11 1
I also know that in general the matrix of a fog (i dont know the word maybe junction or merge?) is the matrix of f multiplied with the matrix of g.
Now, what i cant understand is how am I going to find what the matrix of fog or gof will look like, because f and g come from different dimension spaces. In other words if F is R^2-->R^3 and G is R^3-->R2 then what is FoG or GoF going to be? I prefer to get an answer to this rather than the actual homework question! Thanks in advance!

P.S again sorry for bad english its quite hard for me to find all those weird terms.
 

Answers and Replies

  • #2
HallsofIvy
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Homework Statement


We have the following linear map:
f:R^2-->R^3 f(x,y)=(x,3x+8y,x+y+11y)
Isn't this just f(x,y)= (x, 3x+8y, x+12y)?

e is the standar basis of R^2 and
a is the standar basis of R^3
Question: Is there a linear map g: R^3-->R^2 that the matrix of (fog, a,a) is the 0 matrix?
Is there a linear map g: R^3-->R^2 that the matrix of (gof, e,e) is the 0 matrix?




Homework Equations





The Attempt at a Solution



First of all the matrix of f for the standar basis is:
1 0
Α= 3 8
11 1
With "y+ 11y= 12y", the correct matrix is
[tex]A= \begin{pmatrix}1 & 0 \\ 3 & 8 \\ 1 & 12\end{pmatrix}[/tex]

[tex]\begin{pmatrix}1 & 0 \\ 3 & 8 \\ 11 & 1\end{pmatrix}[/tex]
gives the linear transformation f(x,y)= (x, 3x+ 8y, 11x+ y).

I also know that in general the matrix of a fog (i dont know the word maybe junction or merge?)
The word, in English, is "composition".

is the matrix of f multiplied with the matrix of g.
Now, what i cant understand is how am I going to find what the matrix of fog or gof will look like, because f and g come from different dimension spaces. In other words if F is R^2-->R^3 and G is R^3-->R2 then what is FoG or GoF going to be?
A matrix representing a linear transformation from R3 to R2 must have two rows and three columns. It must be of the form
[tex]B= \begin{pmatrix}a & b & c \\ d & e & f\end{pmatrix}[/tex]
and
[tex]BA= \begin{pmatrix}a & b & c \\ d & e & f\end{pmatrix}\begin{pmatrix}1 & 0 \\ 3 & 8 \\ 1 & 12\end{pmatrix}[/tex]

A matrix representing a linear transformation from R3 to R2 must have three columns and two rows. It must be of the form
[tex]B= \begin{pmatrix} a & b & c \\ d & e & f\end{pmatrix}[/tex]
and
[tex]AB= \begin{pmatrix}1 & 0 \\ 3 & 8 \\ 1 & 12\end{pmatrix}\begin{pmatrix} a & b & c \\ d & e & f\end{pmatrix}[/tex]


I prefer to get an answer to this rather than the actual homework question! Thanks in advance!

P.S again sorry for bad english its quite hard for me to find all those weird terms.
Except for the fact that the word "standard" has a "d" on the end, your English is excellent. Far better than my (put whatever language you please here).
 
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