Linear Transformation

• May 24th 2008, 12:04 AM
pearlyc
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 (Worried) 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.

• May 24th 2008, 12:15 AM
Isomorphism
Quote:

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

$\displaystyle A = \begin{pmatrix} 0 & 0 & 0 \\ 2 & 1 & 0 \end{pmatrix}$
• May 24th 2008, 12:24 AM
Isomorphism
Quote:

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)...

Quote:

3. Whats the meaning of surjective & injective?
Surjective = ONTO mapping (if you know this term)
Injective = ONE-ONE mapping (if you know this term)
• May 24th 2008, 12:36 AM
Isomorphism
Quote:

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 (Worried) 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
$\displaystyle (a_0 + a_1 + 4a_2, 2a_1 + 4a_2 , 4a_2)$.
• May 24th 2008, 06:28 AM
pearlyc
Quote:

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)?
• May 24th 2008, 06:29 AM
pearlyc
Quote:

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) ?
• May 24th 2008, 08:46 AM
Isomorphism
Quote:

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 (Clapping)
• May 24th 2008, 09:22 PM
pearlyc
Quote:

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.
• May 24th 2008, 09:35 PM
Isomorphism
Quote:

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.

Quote:

And the 3 x 3 matrix from the post before this is [T]s right?
Yes
• May 24th 2008, 11:39 PM
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 :(

For example,

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

Show that T is a linear transformation.
• May 25th 2008, 12:14 AM
Isomorphism
Quote:

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 :(