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What's the notation of `[T]` in linear transformation?`T:RR^3->RR^3` is defined by...

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xxxxxyyyya9 | eNotes Newbie

Posted May 15, 2012 at 5:14 AM via web

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What's the notation of `[T]` in linear transformation?

`T:RR^3->RR^3` is defined by `T(x,y,z)=(x+y,2y+x,y+z)`

Find `[T]`

I looked through this chapter, there's no such notation. So I guess it means the standard matrix for `T`, but I'm not sure.

Please tell me if you know this notation. Thanks.

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txmedteach | High School Teacher | (Level 3) Associate Educator

Posted May 31, 2012 at 9:01 PM (Answer #1)

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You're correct. You're looking for the characteristic matrix of the transformation. Thankfully, your transformation makes the process somewhat simple. Let's take care of it in vector form:

`vecx = [[x],[y],[z]]`

Let's also define `T`:

`T = [[t_11, t_12, t_13],[t_21, t_22, t_23],[t_31,t_32,t_33]]`

We can write a matrix equation in the following way:

`Tvecx = [[x+y],[x+2y],[y+z]]`

We now have a complete matrix equation:

`[[t_11, t_12, t_13],[t_21, t_22, t_23],[t_31,t_32,t_33]] [[x],[y],[z]] = [[x+y],[x+2y],[y+z]]`

We have a system of equations from the matrix multiplication:

`t_11x + t_12y + t_13z = x+y`

`t_21x + t_22y + t_23z = x+2y`

`t_31x + t_32y + t_33z = y+z`

Thankfully, there is not much more to solve here. The expressions on the right give us exactly the coefficients on the left. We can now easily solve for `[T]` by copying the numbers on the right in the following way:

`T = [[1,1,0],[1,2,0],[0,1,1]]`

Sources:

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