Linear Algebra - Standard Matrix of T

1. Jun 11, 2017

SetepenSeth

1. The problem statement, all variables and given/known data

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)**

2. Relevant equations

T(x)= Ax

Ax=b

3. The attempt at a solution

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.

2. Jun 11, 2017

Orodruin

Staff Emeritus
Try writing the quantities you are interested in in terms of linear combinations of the quantities that you have been given.

3. Jun 11, 2017

Staff: Mentor

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. Jun 11, 2017

WWGD

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. Jun 17, 2017

SetepenSeth

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. Jun 17, 2017

Ray Vickson

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.