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What's the notation of `[T]` in linear transformation?`T:RR^3->RR^3` is defined by...
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)`
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.
1 Answer | add yours
High School Teacher
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]]`
Posted by txmedteach on May 31, 2012 at 9:01 PM (Answer #1)
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