Key to "Is the transformation Rπ3(x) is linear transformation? Justify..."

Alexander Day

Answered question

2022-02-13

Is the transformation

R_{\frac{π}{3}} (x)

is linear transformation? Justify your answer.

ithangesf4

Beginner2022-02-14Added 16 answers

Step 1
A linear transformation (or a linear map) is a function

T : R^{n} \to R^{m}

that satisfies the following properties:
1)

T (x + y) = T (x) + T (y)

T (a x) = a T (x)

for any vectors

x, y \in R^{n}

and any scalar

a \in R

Now, since we have

R_{θ} = [\begin{array}{ccc} \cos θ & - \sin θ & 0 \\ \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{array}]

then

R_{θ} (x) = [\begin{array}{ccc} \cos \frac{π}{3} & - \sin \frac{π}{3} & 0 \\ \sin \frac{π}{3} & \cos \frac{π}{3} & 0 \\ 0 & 0 & 1 \end{array}] = [\begin{array}{ccc} \frac{1}{2} & - \frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 1 \end{array}]

Now, since every matrix is a linear transformation. Hence the given transformation is a linear transformtion.

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?