Step by step solution to "Whether the function is a linear transformation or..."

allhvasstH

Answered question

2021-01-08

Whether the function is a linear transformation or not.

T : R^{2} \to R^{2}, T (x, y) = (x, y^{2})

okomgcae

Skilled2021-01-09Added 93 answers

Calculation:
The function is defined as,

T (x, y) = (x, y^{2})

Assume two general vectors

u = (u_{1}, u_{2})

and

v = (v_{1}, v_{2})

Then

u + v = (u_{1} + v_{1}, u_{2} + v_{2})

c u = (c u_{1}, c u_{2})

The function is a linear transformation if it satisfies the two properties as mentioned in the approach part.
Compute

T (u + v)

and

T (u) + T (v)

T (u + v) = T (u_{1} + v_{1}, u_{2} + v_{2})

= (u_{1} + v_{1}, {(u_{2} + v_{2})}^{2})

= (u_{1} + v_{1}, u_{2}^{2} + v_{2}^{2} + 2 u_{2} v_{2})

T (u) + T (v) = T (u_{1}, u_{2}) + T (v_{1}, v_{2})

= (u_{1}, u_{2}^{2}) + (v_{1}, v_{2}^{2})

= (u_{1} + v_{1}, u_{2}^{2} + v_{2}^{2})

Since

T (u + v) \neq q T (u) + T (v)

, the first property is not satisfied.
Therefore, the function is not a linear transformation.

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?