Linear algebra, problem with linear maps and their matrices

You could have fooled me into thinking that you were a native speaker of English! In summary, the conversation discusses a linear map with a specific matrix and the question of whether there is another linear map that would result in a zero matrix when composed with the original map. The conversation also touches on finding the matrix of composition for maps with different dimension spaces, and concludes with a request for an explanation rather than a solution to the homework question. The expert also notes that the speaker's English is excellent.
  • #1
sphlanx
11
0

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 don't know the word maybe junction or merge?) is the matrix of f multiplied with the matrix of g.
Now, what i can't 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.
 
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  • #2
sphlanx said:

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 don't 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 can't 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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  • #3


I would approach this problem by first understanding the concept of linear maps and their matrices. Linear maps are functions that preserve vector addition and scalar multiplication, and their matrices represent the transformation of vectors from one space to another.

In this case, we are given a linear map f:R^2-->R^3 and asked to find a linear map g:R^3-->R^2 that results in a zero matrix when composed with f. This means that the resulting matrix of (fog, a,a) will be a 3x2 matrix with all elements equal to zero. Similarly, for the second question, we need to find a g:R^3-->R^2 that results in a zero matrix when composed with f, this time with the matrix of (gof, e,e).

To find such a g, we can use the concept of inverse matrices. The inverse of a matrix A is denoted as A^-1 and is defined as the matrix that, when multiplied with A, results in the identity matrix. In other words, A^-1 * A = I where I is the identity matrix.

Using this concept, we can find the inverse of the matrix A that represents f for the standard basis. Once we have the inverse, we can compose it with f to get a zero matrix. This is because (A^-1 * A) * v = A^-1 * (A * v) = A^-1 * 0 = 0, where v is any vector in R^2.

Similarly, for the second question, we can find the inverse of A and compose it with f to get a zero matrix. This time, we will use the matrix A that represents f for the standard basis of R^3.

In conclusion, the linear map g that we are looking for is the inverse of the matrix A that represents f for the standard basis. This will result in a zero matrix when composed with f, satisfying both of the given conditions.
 

1. What is linear algebra and why is it important?

Linear algebra is a branch of mathematics that deals with linear equations and their representations through vector spaces and matrices. It is important because it provides a powerful framework for solving real-world problems in fields such as physics, engineering, and computer science.

2. What are linear maps and how are they related to matrices?

Linear maps, also known as linear transformations, are functions that preserve vector addition and scalar multiplication. They are related to matrices because they can be represented by matrices, and the properties of linear maps can be studied through the associated matrices.

3. What are the common problems encountered in linear algebra?

Some common problems in linear algebra include finding solutions to systems of linear equations, computing determinants and inverses of matrices, and diagonalizing matrices. Other challenges may arise when working with high-dimensional data or dealing with ill-conditioned matrices.

4. How can linear algebra be applied in real-world situations?

Linear algebra has numerous applications in fields such as physics, engineering, and computer science. It is used to model and solve problems involving systems of linear equations, optimization, data analysis, and image processing. It also serves as a foundation for more advanced mathematical concepts.

5. How can one improve their understanding of linear algebra?

One can improve their understanding of linear algebra by practicing problems, working through proofs, and studying the properties and applications of matrices and linear transformations. It is also helpful to familiarize oneself with relevant software tools, such as MATLAB or Python, for performing computations and visualizing concepts.

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