Hi. I need help understanding this question:

The question tells us to state why the following is not a linear transformation:

\(\displaystyle T: R^2 \rightarrow R^2\)

\(\displaystyle T(x,y) = (x^2, xy)\)

Their answer is simple enough:

There's a property: T(-v) = -T(v)

however, T(-(1,1) ǂ -T(1,1) \(\displaystyle \Longrightarrow\) (by the way, is there a way to post the not equal sign in Latex. I just copy pasted from character map)

therefore, it is not a linear transformation.

But I am having trouble solving the question using the main two properties:

T(a+b) = T(a) + T(b)

T(\(\displaystyle \lambda a\)) = \(\displaystyle \lambda T(a)\)

I started like this:

T(x,y) = (x^2, xy)

T(a+b) = T(\(\displaystyle \alpha_1 + \beta_1, \alpha_2 + \beta_2\))=

=(\(\displaystyle (\alpha_1 + \beta_1)^2\), \(\displaystyle (\alpha_1 + \beta_1)(\alpha_2 + \beta_2)\))=

= (\(\displaystyle \alpha_1^2 + 2\alpha_1\beta_1 + \beta_1^2\)), (\(\displaystyle \alpha_1\alpha_2 + \alpha_1\beta_2 + \beta_1\alpha_2 + \beta_1\beta_2\))

and from here I am having trouble proving or disproving if it equals T(a) + T(b).

Can someone help?

Thanks.

The question tells us to state why the following is not a linear transformation:

\(\displaystyle T: R^2 \rightarrow R^2\)

\(\displaystyle T(x,y) = (x^2, xy)\)

Their answer is simple enough:

There's a property: T(-v) = -T(v)

however, T(-(1,1) ǂ -T(1,1) \(\displaystyle \Longrightarrow\) (by the way, is there a way to post the not equal sign in Latex. I just copy pasted from character map)

therefore, it is not a linear transformation.

But I am having trouble solving the question using the main two properties:

T(a+b) = T(a) + T(b)

T(\(\displaystyle \lambda a\)) = \(\displaystyle \lambda T(a)\)

I started like this:

T(x,y) = (x^2, xy)

T(a+b) = T(\(\displaystyle \alpha_1 + \beta_1, \alpha_2 + \beta_2\))=

=(\(\displaystyle (\alpha_1 + \beta_1)^2\), \(\displaystyle (\alpha_1 + \beta_1)(\alpha_2 + \beta_2)\))=

= (\(\displaystyle \alpha_1^2 + 2\alpha_1\beta_1 + \beta_1^2\)), (\(\displaystyle \alpha_1\alpha_2 + \alpha_1\beta_2 + \beta_1\alpha_2 + \beta_1\beta_2\))

and from here I am having trouble proving or disproving if it equals T(a) + T(b).

Can someone help?

Thanks.

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