# Notation used in matrix representation of linear transformation

Hello guys,

Let ##T: \mathbb{R^2} \to \mathbb{R^2}##. Suppose I have standard basis ##B = \{u_1, u_2\}## and another basis ##B^{\prime} = \{v_1, v_2\}## The linear transformation is described say as such ##T(v_1) = v_1 + v_2, T(v_2) = v_1##

If I want to write the matrix representing ##T## with respect to basis ##B^{\prime}## then I'll just find ##[T]_{B'}##. I can also find ##[T]_{B}## rather straightforward using similarity transformation if I know the transition matrix between those two bases.

But suddenly I encounter this notation ##[T]_{B,B'}##. I don't know exactly what this notation represents. Do you guys know what this notation mean? What other matrix should I provide in this case? Normally I use that comma subscript to denote transition matrix between bases, but never for linear transformation matrix.

Thank You

## Answers and Replies

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It means that you take the basis ##B^\prime## on the domain and ##B## on the codomain (both are ##\mathbb{R}^2##). Or the other way around, depending on who is using the notation.

So the idea is to see what happens to ##v_i##. So look at ##T(v_1)## and ##T(v_2)## and express these in the ##\{u_1,u_2\}## basis. So you write ##T(v_1) = \alpha u_1 + \beta u_2## and ##T(v_2) = \gamma u_1 + \delta u_2##. The the matrix you seek is

$$\left(\begin{array}{cc} \alpha & \gamma\\ \beta & \delta \end{array}\right)[tex] It means that you take the basis ##B^\prime## on the domain and ##B## on the codomain (both are ##\mathbb{R}^2##). Or the other way around, depending on who is using the notation. So the idea is to see what happens to ##v_i##. So look at ##T(v_1)## and ##T(v_2)## and express these in the ##\{u_1,u_2\}## basis. So you write ##T(v_1) = \alpha u_1 + \beta u_2## and ##T(v_2) = \gamma u_1 + \delta u_2##. The the matrix you seek is [tex]\left(\begin{array}{cc} \alpha & \gamma\\ \beta & \delta \end{array}\right)$$
Thanks micromass for the help. It makes sense. I managed to get that matrix by post-multiplying ##[T]_b## with the transition matrix ##P_{B' \to B}##. I was just really confused because one of the text that I'm reading apparently got the matrix wrong. (Not considering the fact that they use comma and arrow notation interchangeably)

Thanks micromass for the help. It makes sense. I managed to get that matrix by post-multiplying ##[T]_b## with the transition matrix ##P_{B' \to B}##.
That works too.

I was just really confused because one of the text that I'm reading apparently got the matrix wrong. (Not considering the fact that they use comma and arrow notation interchangeably)
What did the text say?

That works too.

What did the text say?
It's an example problem. The desired matrix is just the same with my own work, but somehow transposed.