Linear Algebra - Standard Matrix of T

In summary, a linear transformation from ℝ² to P² is given with the operations T [1 1] = 1 - 2x and T [3 -1] = x+2x². The goal is to find T [-7 9] and T [a b]. The associated standard matrix is 3x2, which cannot be inverted. To find the matrix, we can use the known values of T [1 1] and T [3 -1] to form a system of three linear equations in six variables. However, this approach may not be successful if the linear transformation is not injective or if the dimension of its image is 1.
  • #1
SetepenSeth
16
0

Homework Statement



Let T: ℝ² → a linear transformation with usual operations such as

T [1 1] = 1 - 2x and
T [3 -1]= x+2x²

Find T [-7 9] and T [a b]

**Though I'm writing them here as 1x 2 row vectors , all T's are actually 2x1 column vectors (I didn't see a way to give them proper format)**

Homework Equations


[/B]
T(x)= Ax

Ax=b

The Attempt at a Solution


[/B]
Finding the transformation for the required vectors is not an issue, as long as I have the associated standard matrix, however my approaches to find this matrix have not come out successful.

Since the transformation goes ℝ² → it is my understanding that the associated matrix is 3x2 so using the inverse or the transformations of the canonical vectors T(ei) don't seem to help much (or I am failing to see how to properly apply them). The other option it is that this is a linear transformation that is not matricial, in which case I'm uncertain on how to approach it.

Any advise would be appreciated.
 
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  • #2
Try writing the quantities you are interested in in terms of linear combinations of the quantities that you have been given.
 
  • #3
SetepenSeth said:

Homework Statement



Let T: ℝ² → a linear transformation with usual operations such as

T [1 1] = 1 - 2x and
T [3 -1]= x+2x²

Find T [-7 9] and T [a b]

**Though I'm writing them here as 1x 2 row vectors , all T's are actually 2x1 column vectors (I didn't see a way to give them proper format)**

Homework Equations


[/B]
T(x)= Ax

Ax=b

The Attempt at a Solution


[/B]
Finding the transformation for the required vectors is not an issue, as long as I have the associated standard matrix, however my approaches to find this matrix have not come out successful.

Since the transformation goes ℝ² → it is my understanding that the associated matrix is 3x2 so using the inverse or the transformations of the canonical vectors T(ei) don't seem to help much (or I am failing to see how to properly apply them). The other option it is that this is a linear transformation that is not matricial, in which case I'm uncertain on how to approach it.

Any advise would be appreciated.
I'm not really sure if I understood you correctly. First for the easy part, you can have a look here on how to use LaTex code to write vectors and matrices. As you've observed ##\mathbb{R}^2## is two dimensional and ##P^2## three dimensional which makes any matrix representations of linear mappings ##T \, : \,\mathbb{R}^2 \longrightarrow P^2## a ##3 \times 2## matrix which can't be inverted. In order to find a matrix ##A_T## for ##T## we have to find six numbers ##A_T = \begin{bmatrix}x_{00} & x_{01} \\ x_{10} & x_{11} \\ x_{20} & x_{21} \end{bmatrix}## which are arbitrary as long as we do not require any conditions on them. This is the basis of the next considerations and what you have correctly said as well.

Then you know some vectors ##\vec{a} = \begin{bmatrix}a_0 \\ a_1\end{bmatrix} \in \mathbb{R}^2## for which you know what ##T(\vec{a})## is, say according to a basis ##\{1,x,x^2\} \subset P^2##. This means you know ##T(\vec{a}) = \vec{b} = b_0 \cdot 1 + b_1 \cdot x + b_2 \cdot x^2##.
If nothing else is told ##\vec{a} = a_0 \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix}+a_1 \cdot \begin{bmatrix}0 \\ 1 \end{bmatrix}##.

So now you can simply multiply ##A_T \cdot \vec{a} = \vec{b}## which gives you a system of three linear equations in six variables. In the examples above this is underdeterminated, which depends on how many equations ##T(\vec{a}) = \vec{b}## you have.

As I wasn't really able to follow you, I'm uncertain if this answers your question. And I don't know what "matricial" means. In the end all depends on how ##T## is defined. It can be an embedding (injective), ##T=0## is a possibility, and ##\operatorname{dim} \operatorname{im} T = 1## is a (the) third option.
 
  • #4
SetepenSeth said:

Homework Statement



Let T: ℝ² → a linear transformation with usual operations such as

T [1 1] = 1 - 2x and
T [3 -1]= x+2x²

Find T [-7 9] and T [a b]

**Though I'm writing them here as 1x 2 row vectors , all T's are actually 2x1 column vectors (I didn't see a way to give them proper format)**

Homework Equations


[/B]
T(x)= Ax

Ax=b

The Attempt at a Solution


[/B]
Finding the transformation for the required vectors is not an issue, as long as I have the associated standard matrix, however my approaches to find this matrix have not come out successful.

Since the transformation goes ℝ² → it is my understanding that the associated matrix is 3x2 so using the inverse or the transformations of the canonical vectors T(ei) don't seem to help much (or I am failing to see how to properly apply them). The other option it is that this is a linear transformation that is not matricial, in which case I'm uncertain on how to approach it.

Any advise would be appreciated.

By linearity of the map, if {(a,b),(c,d)} is a basis, then ## (e,f)=k_1(a,b)+k_2(c,d) ## then ##T(e,f)=T(k_1(a,b))+ T(k_2(c,d)) ##

For the other part, do what Orodruin suggested: express T(v) as a linear combination in terms of basis vectors in ## P_2 ##. Notice how nice and conveninet are the "natural" bases ##e_1=(1,0), e_2=(0,1) ## and their equivalents in higher dmension.
 
  • #5
fresh_42 said:
I'm not really sure if I understood you correctly. First for the easy part, you can have a look here on how to use LaTex code to write vectors and matrices. As you've observed ##\mathbb{R}^2## is two dimensional and ##P^2## three dimensional which makes any matrix representations of linear mappings ##T \, : \,\mathbb{R}^2 \longrightarrow P^2## a ##3 \times 2## matrix which can't be inverted. In order to find a matrix ##A_T## for ##T## we have to find six numbers ##A_T = \begin{bmatrix}x_{00} & x_{01} \\ x_{10} & x_{11} \\ x_{20} & x_{21} \end{bmatrix}## which are arbitrary as long as we do not require any conditions on them. This is the basis of the next considerations and what you have correctly said as well.

Then you know some vectors ##\vec{a} = \begin{bmatrix}a_0 \\ a_1\end{bmatrix} \in \mathbb{R}^2## for which you know what ##T(\vec{a})## is, say according to a basis ##\{1,x,x^2\} \subset P^2##. This means you know ##T(\vec{a}) = \vec{b} = b_0 \cdot 1 + b_1 \cdot x + b_2 \cdot x^2##.
If nothing else is told ##\vec{a} = a_0 \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix}+a_1 \cdot \begin{bmatrix}0 \\ 1 \end{bmatrix}##.

So now you can simply multiply ##A_T \cdot \vec{a} = \vec{b}## which gives you a system of three linear equations in six variables. In the examples above this is underdeterminated, which depends on how many equations ##T(\vec{a}) = \vec{b}## you have.

As I wasn't really able to follow you, I'm uncertain if this answers your question. And I don't know what "matricial" means. In the end all depends on how ##T## is defined. It can be an embedding (injective), ##T=0## is a possibility, and ##\operatorname{dim} \operatorname{im} T = 1## is a (the) third option.

Thanks for the advise. I didn't know how to use LaTeX (how do you pronounce it? though).

I get to the part where I have 6 variables to express, however I'm still uncertain on how do I go into finding this variables if I only have two ##T(\vec{a}) = \vec{b}## to work with, I'm pretty sure this would have come out just painless with just one more ##T## to work with.

By the way, "linear transformation that is not matricial" I meant a transformation that is not a matrix transformation (bad habit of mixing languages, sorry).
 
  • #6
SetepenSeth said:

Homework Statement



Let T: ℝ² → a linear transformation with usual operations such as

T [1 1] = 1 - 2x and
T [3 -1]= x+2x²

Find T [-7 9] and T [a b]

**Though I'm writing them here as 1x 2 row vectors , all T's are actually 2x1 column vectors (I didn't see a way to give them proper format)**

Homework Equations


[/B]
T(x)= Ax

Ax=b

The Attempt at a Solution


[/B]
Finding the transformation for the required vectors is not an issue, as long as I have the associated standard matrix, however my approaches to find this matrix have not come out successful.

Since the transformation goes ℝ² → it is my understanding that the associated matrix is 3x2 so using the inverse or the transformations of the canonical vectors T(ei) don't seem to help much (or I am failing to see how to properly apply them). The other option it is that this is a linear transformation that is not matricial, in which case I'm uncertain on how to approach it.

Any advise would be appreciated.

It does not matter whether you write vectors as row or column vectors, as long as you don't bother using matrices (which only get in the way in this problem). Just find out how to write e1 = [1 0] and e2 = [0 1] as linear combinations of v1 = [1 1] and v2 = [3 -1] (a very easy task).

So, if e1 = a1 v1 + a2 v2 and e2 = b1 v1 + b2 v2, then you can get T(e1) as a1 T(v1) + a2 T(v2), etc. Then write [-7 9] in terms of e1 and e2, etc, and apply linearity of T.

It is all elementary, and no matrices need be involved at all. As I said, matrices just get in the way.
 

Related to Linear Algebra - Standard Matrix of T

1. What is the standard matrix of a linear transformation?

The standard matrix of a linear transformation is a matrix that represents the transformation's actions on the standard basis vectors of a vector space. It is used to perform calculations and transformations on vectors in a more efficient manner.

2. How is the standard matrix of a linear transformation determined?

The standard matrix of a linear transformation is determined by applying the transformation to each of the standard basis vectors and recording the resulting coordinates. These coordinates are then arranged in a matrix, with each column representing the transformation's action on a specific basis vector.

3. Can the standard matrix of a linear transformation be used to perform transformations on other vectors?

Yes, the standard matrix of a linear transformation can be used to perform transformations on any vector in the same vector space. This is because all vectors in the vector space can be expressed as a linear combination of the standard basis vectors, and the standard matrix represents the transformation's actions on these basis vectors.

4. What is the relationship between the standard matrix of a linear transformation and the transformation's properties?

The standard matrix of a linear transformation contains important information about the transformation's properties. For example, the number of columns in the matrix represents the dimension of the vector space, and the rank of the matrix corresponds to the dimension of the transformation's range. The matrix can also be used to determine if the transformation is invertible or if it has any eigenvalues.

5. Can the standard matrix of a linear transformation change depending on the basis vectors used?

Yes, the standard matrix of a linear transformation can change depending on the basis vectors used. This is because the matrix is constructed based on the transformation's actions on the standard basis vectors, and different basis vectors will result in different coordinates and therefore a different matrix. However, the properties of the transformation, such as invertibility and eigenvalues, will remain the same regardless of the basis vectors used.

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