Linear Algebra - Show that T is Linear

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.

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

The Attempt at a Solution

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Mark44
Mentor

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.

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.

would T(cx+da) be something like:

cx1 + da1.

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

cx3 + da3.

Office_Shredder
Staff Emeritus
Gold Member
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)

Mark44
Mentor
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
Staff Emeritus
Gold Member
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.