1. Linear Transformation

Hey guys,

I was revising Linear Transformation and I came accross a couple of doubts here and there. Hope you guys would be able to explain to me for me to understand it better (:

1. I know how to attempt questions with two variables, eg. S(x,y) = (2x-y, x+y) to prove that they are linear transformations. What is the method to attempt those with three variables involved? eg. T(x,y,z) = (0, 2x+y) or F(x,y,z) = $\displaystyle {\color{white}.} \quad \left[ \begin{array}{ccc}y & z\\ -x & 0 \end{array}\right]$

2. What is the way to see the transformation patterns ? eg. $\displaystyle {\color{white}.} \quad \left[ \begin{array}{ccc}0 & 1\\ 1 & 0 \end{array}\right]$ is the reflection of y = x. Say, how do we determine what is the transformation for a matrix like $\displaystyle {\color{white}.} \quad \left[ \begin{array}{ccc}0 & 0\\ 1 & 0 \end{array}\right]$ or $\displaystyle {\color{white}.} \quad \left[ \begin{array}{ccc}1 & 0\\ a & 1 \end{array}\right]$, etc.

3. Whats the meaning of surjective & injective?

4. I was taught how to tackle most of the questions in matrix form. When it comes to polynomial, I'd go nuts figuring how to attempt the question Especially when it comes to change of basis questions. Like what is [T]s and things like that when from matrix you can find it as the matrix representation.

eg. T[p(x)] = p(2x+1), B = {1, x, x^2}

$\displaystyle T (a_0 + a_1x + a_2x^2) = a_0 + a_1(2x+1) + a_2(2x+1)^2$

Find [T]B.

2. Originally Posted by pearlyc
1. I know how to attempt questions with two variables, eg. S(x,y) = (2x-y, x+y) to prove that they are linear transformations. What is the method to attempt those with three variables involved? eg. T(x,y,z) = (0, 2x+y) or F(x,y,z) = $\displaystyle {\color{white}.} \quad \left[ \begin{array}{ccc}y & z\\ -x & 0 \end{array}\right]$

How different is a three variable one?

If T(x,y,z) = (0, 2x+y) is a linear transformation, then you can find a matrix A such that this A transforms (x,y,z) to (0,2x+y). That is

$\displaystyle A(x,y,z)^T = (0,2x+y)^T$
So the matrix should map a 3 x 1 vector to a 2 x 1 vector. This means A must be 2 x 3.

So assume A to be a general 2 x 3 matrix and solve for it to get A as

$\displaystyle A = \begin{pmatrix} 0 & 0 & 0 \\ 2 & 1 & 0 \end{pmatrix}$

3. Originally Posted by pearlyc
Hey guys,

2. What is the way to see the transformation patterns ? eg. $\displaystyle {\color{white}.} \quad \left[ \begin{array}{ccc}0 & 1\\ 1 & 0 \end{array}\right]$ is the reflection of y = x. Say, how do we determine what is the transformation for a matrix like $\displaystyle {\color{white}.} \quad \left[ \begin{array}{ccc}0 & 0\\ 1 & 0 \end{array}\right]$ or $\displaystyle {\color{white}.} \quad \left[ \begin{array}{ccc}1 & 0\\ a & 1 \end{array}\right]$, etc.
This is simple.. Assume (x,y) to be the mapping vector and see what the mapped vector is

For $\displaystyle \quad \left[ \begin{array}{ccc}0 & 1\\ 1 & 0 \end{array}\right]$
$\displaystyle \quad \left[ \begin{array}{ccc}0 & 1\\ 1 & 0 \end{array}\right] \cdot \begin{pmatrix} x\\ y\end{pmatrix} = \begin{pmatrix} y\\ x\end{pmatrix}$

So this transformation maps all (x,y) to (y,x). Thus its the reflection

For $\displaystyle \quad \left[ \begin{array}{ccc}1 & 0\\ a & 1 \end{array}\right]$

Do the same thing again to see that it maps (x,y) to (x,ax+y)...

3. Whats the meaning of surjective & injective?
Surjective = ONTO mapping (if you know this term)
Injective = ONE-ONE mapping (if you know this term)

4. Originally Posted by pearlyc
4. I was taught how to tackle most of the questions in matrix form. When it comes to polynomial, I'd go nuts figuring how to attempt the question Especially when it comes to change of basis questions. Like what is [T]s and things like that when from matrix you can find it as the matrix representation.

eg. T[p(x)] = p(2x+1), B = {1, x, x^2}

$\displaystyle T (a_0 + a_1x + a_2x^2) = a_0 + a_1(2x+1) + a_2(2x+1)^2$

Find [T]B.
Dont worry about it just because its a polynomial. After all $\displaystyle a_0 + a_1x + a_2x^2$ can be viewed as the tuple $\displaystyle (a_0, a_1, a_2)$ with B as basis. With this look it is just another $\displaystyle \mathbb{R}^3$ vector, isnt it?

So the transformation looks like $\displaystyle T[p(x)] = p(2x+1)$ and we want to find the matrix representation for this, right?

$\displaystyle T (a_0 + a_1x + a_2x^2) = a_0 + a_1(2x+1) + a_2(2x+1)^2$
$\displaystyle a_0 + a_1(2x+1) + a_2(2x+1)^2 = a_0 + 2a_1x+a_1 + 4a_2 x^2 + 4 a_2 x +a_2$
$\displaystyle a_0 + 2a_1x+a_1 + 4a_2 x^2 + 4 a_2 x +a_2 = (a_0 + a_1 + 4a_2) +(2a_1 + 4a_2)x + (4a_2)x^2$

So $\displaystyle a_0 + a_1x + a_2x^2$ gets mapped to $\displaystyle (a_0 + a_1 + 4a_2) +(2a_1 + 4a_2)x + (4a_2)x^2$.

In terms of triples, its the same as saying $\displaystyle (a_0, a_1, a_2)$ maps to
$\displaystyle (a_0 + a_1 + 4a_2, 2a_1 + 4a_2 , 4a_2)$

So now this problem is the same as the first question

Just a\find a 3 x 3 matrix that maps $\displaystyle (a_0, a_1, a_2)$ maps to
$\displaystyle (a_0 + a_1 + 4a_2, 2a_1 + 4a_2 , 4a_2)$.

5. Originally Posted by Isomorphism
How different is a three variable one?

If T(x,y,z) = (0, 2x+y) is a linear transformation, then you can find a matrix A such that this A transforms (x,y,z) to (0,2x+y). That is

$\displaystyle A(x,y,z)^T = (0,2x+y)^T$
So the matrix should map a 3 x 1 vector to a 2 x 1 vector. This means A must be 2 x 3.

So assume A to be a general 2 x 3 matrix and solve for it to get A as

$\displaystyle A = \begin{pmatrix} 0 & 0 & 0 \\ 2 & 1 & 0 \end{pmatrix}$
How do I do the whole proving the two properties :

T(u+v) = T(u) + T(v) and T(au) = aT(u)?

6. Originally Posted by Isomorphism
Dont worry about it just because its a polynomial. After all $\displaystyle a_0 + a_1x + a_2x^2$ can be viewed as the tuple $\displaystyle (a_0, a_1, a_2)$ with B as basis. With this look it is just another $\displaystyle \mathbb{R}^3$ vector, isnt it?

So the transformation looks like $\displaystyle T[p(x)] = p(2x+1)$ and we want to find the matrix representation for this, right?

$\displaystyle T (a_0 + a_1x + a_2x^2) = a_0 + a_1(2x+1) + a_2(2x+1)^2$
$\displaystyle a_0 + a_1(2x+1) + a_2(2x+1)^2 = a_0 + 2a_1x+a_1 + 4a_2 x^2 + 4 a_2 x +a_2$
$\displaystyle a_0 + 2a_1x+a_1 + 4a_2 x^2 + 4 a_2 x +a_2 = (a_0 + a_1 + 4a_2) +(2a_1 + 4a_2)x + (4a_2)x^2$

So $\displaystyle a_0 + a_1x + a_2x^2$ gets mapped to $\displaystyle (a_0 + a_1 + 4a_2) +(2a_1 + 4a_2)x + (4a_2)x^2$.

In terms of triples, its the same as saying $\displaystyle (a_0, a_1, a_2)$ maps to
$\displaystyle (a_0 + a_1 + 4a_2, 2a_1 + 4a_2 , 4a_2)$

So now this problem is the same as the first question

Just a\find a 3 x 3 matrix that maps $\displaystyle (a_0, a_1, a_2)$ maps to
$\displaystyle (a_0 + a_1 + 4a_2, 2a_1 + 4a_2 , 4a_2)$.
Does this mean the 3 x 3 matrix for that is
(1 1 4
0 2 4
0 0 4) ?

7. Originally Posted by pearlyc
Does this mean the 3 x 3 matrix for that is
(1 1 4
0 2 4
0 0 4) ?
Bingo

8. Originally Posted by pearlyc
How do I do the whole proving the two properties :

T(u+v) = T(u) + T(v) and T(au) = aT(u)?

And the 3 x 3 matrix from the post before this is [T]s right? (:

Thanks a lot for your help, really.

9. Originally Posted by pearlyc
Once you find a matrix A such that $\displaystyle T(x) = Ax$, there is nothing to prove since you must have proved the map Ax is linear for any matrix A.

But if you are still adamant, you can try the axioms directly.

And the 3 x 3 matrix from the post before this is [T]s right?
Yes

10. Thanks a lot.

Okay so if you have [T]s, to find [T]b you'll need Pb->s and Ps->b right? How do we find this from a polynomial function? Let's take the question above as an example?

How do you prove whether a polynomial function is a linear transformation then?

I am still so weak in this whole polynomial business

For example,

T[p(x)] = xp(x)
T(a + bx + cx^2) = ax + bx^2 + cx^3

Show that T is a linear transformation.

11. Originally Posted by pearlyc
Thanks a lot.

Okay so if you have [T]s, to find [T]b you'll need Pb->s and Ps->b right? How do we find this from a polynomial function? Let's take the question above as an example?

How do you prove whether a polynomial function is a linear transformation then?

I am still so weak in this whole polynomial business