Linear transformations and standart matrices

[PirateFan308]

Homework Statement

Define the linear transformation $T: R^{3} → R^{3}$ by $T(v)=$ the projection of $v$ onto the vector $w=(1,2,1)$

Find the (standard matrix of $T$)

Homework Equations

$T: V → W$ is a function from V to W (which means that for each v in V, there is a T9v) in W such that:
T(v+v')=T(v)+T(v'), T(cv) = cT(v) for all v,v'$\in$V, c$\in$F

For T(v) = Av, the matrix A is called the standard matrix of T

projection of v onto w = $(\frac{v \bullet w}{w \bullet w})(w)$

The Attempt at a Solution

I'm having problems understanding Linear Transformations at all, and I'm not really sure if this is at all correct ... I'm thinking I should apply T to $e_1, e_2, and~ e_3$.

Because the transformation takes place in $R^{3}$, I know A will be a 3x3 matrix. Apply T to $e_1$, where proj $e_1$ onto w would be $(1/6, 2/6, 1/6)$ and we can see that $x=1/6, y=1/3, z=1/6$, so should the first column of A be $\begin{pmatrix} 1/6 \\ 1/3 \\ 1/6 \end{pmatrix}$? Then apply T to $e_2$, where proj $e_2$ onto w would be $(1/3, 2/3, 1/3)$ and we can see that $x=1/3, y=2/3, z=1/3$, so the second column of A should be $\begin{pmatrix} 1/3 \\ 2/3 \\ 1/3 \end{pmatrix}$. And again, applying T to $e_3$ proj $e_3$ onto w would be $(1/6, 2/6, 1/6)$ and we can see that $x=1/6, y=1/3, z=1/6$, so should the third column of A be $\begin{pmatrix} 1/6 \\ 1/3 \\ 1/6 \end{pmatrix}$?

So should the matrix A be $A=\begin{pmatrix} 1/6 & 1/3 & 1/6 \\ 1/3 & 2/3 & 1/6 \\ 1/6 & 1/3 & 1/6 \end{pmatrix}$? This seems odd to have the third row be the same as the first row and the third column the same as the third column.

Is this correct, or am I completely off? Thanks!

 

[mighty2000]

Dear fellow scientist,

Your attempt at finding the standard matrix of T is on the right track, but there are a few errors in your calculations. Let's work through it together to find the correct answer.

First, let's review the definition of a linear transformation. It is a function that takes vectors from one vector space and maps them to another vector space, while preserving certain properties such as linearity (as stated in the homework equations). In this case, T is a linear transformation from R^3 to R^3.

Next, we need to understand what the projection of a vector onto another vector means. The projection of a vector v onto a vector w is the component of v that lies in the direction of w. In other words, it is the vector that is parallel to w and has the same magnitude as the component of v in the direction of w. This can be calculated using the formula you provided: proj_v onto w = (v dot w / w dot w) * w.

Now, let's apply this to find the standard matrix of T. We need to find the images of the standard basis vectors e_1, e_2, and e_3 under T. The standard basis vectors are the unit vectors along the x, y, and z axes respectively.

Applying T to e_1, we get proj_e_1 onto w = (1 dot 1 + 0 dot 2 + 0 dot 1) / (1 dot 1 + 2 dot 2 + 1 dot 1) * (1, 2, 1) = (2/6, 4/6, 2/6) = (1/3, 2/3, 1/3). This is the first column of the standard matrix.

Similarly, applying T to e_2, we get proj_e_2 onto w = (0 dot 1 + 1 dot 2 + 0 dot 1) / (1 dot 1 + 2 dot 2 + 1 dot 1) * (1, 2, 1) = (2/6, 4/6, 2/6) = (1/3, 2/3, 1/3). This is the second column of the standard matrix.

Finally, applying T to e_3, we get proj_e_3 onto w = (0 dot 1 +