Linear Algebra - Show that T is Linear

[1LastTry]

Homework Statement

Let y $\in$ℝ$^{3}$ be a fixed vector, and define T:ℝ$^{3}$→ℝ$^{3}$ to be Tx = X $\times$ Y, the cross product of x and y.
Show that T is linear.

Homework Equations

The Attempt at a Solution

For this question do we have to define another T with the cross product of two other variables to prove this?

For example:

Tx = x $\times$ y
Ta = a $\times$ b

and we can prove linearity by T(cx+da) = cTx + dTa ?

Solution in this problem is welcome, since this assignment is already overdue.

Thanks

Homework Statement

Homework Equations

The Attempt at a Solution

 

[Mark44]

Homework Statement

Let y $\in$ℝ$^{3}$ be a fixed vector, and define T:ℝ$^{3}$→ℝ$^{3}$ to be Tx = X $\times$ Y, the cross product of x and y.
Show that T is linear.

Homework Equations

The Attempt at a Solution

For this question do we have to define another T with the cross product of two other variables to prove this?

For example:

Tx = x $\times$ y
Ta = a $\times$ b

No, Ta = a X y

and we can prove linearity by T(cx+da) = cTx + dTa ?

Yes.

Solution in this problem is welcome, since this assignment is already overdue.

On-time or overdue, PF policy is that we don't do the work for you. We'll help you with it, though.

 

[1LastTry]

would T(cx+da) be something like:

cx1 + da1.

cx2 + da2. x(cross product) [y1,y2,y3]?

cx3 + da3.

 

[Office_Shredder]

Assuming that you have a column vector on the left and a row vector on the right, that is exactly what it would look like.

You can save yourself a bit of notational pain by doing this in two steps: T is linear if T(x+a) = T(x) + T(a) for all vectors x and a, and T(cx) = cT(x) for all vectors x and scalars c. This way you don't have to worry about keeping track of c's and d's when doing the hard part (showing additive linearity)

 

[1LastTry]

Thanks for your helps.

 

[Mark44]

Assuming that you have a column vector on the left and a row vector on the right, that is exactly what it would look like.

Office_Shredder, I think you have a misconception here. This transformation performs the cross product (not matrix product) of its argument and some fixed vector in R³. For the ordinary cross product, all you need are two vectors in R³.

You can save yourself a bit of notational pain by doing this in two steps: T is linear if T(x+a) = T(x) + T(a) for all vectors x and a, and T(cx) = cT(x) for all vectors x and scalars c. This way you don't have to worry about keeping track of c's and d's when doing the hard part (showing additive linearity)

 

[Office_Shredder]

OK I agree there was no reason to write it in that particular format, I simply meant that it wasn't 100% clear to me if he was trying to write two vectors being cross-producted with each other.