Finding the Standard Matrix for Linear Transformation

Question

We are asked to find the standard matrix $A$ for $T$:

Consider the transformation $T : \mathbb{R^3} \rightarrow \mathbb{R^4}$ given by

$$T(x_1, x_2, x_3) = (x_1 + x_2 + x_3, x_2 + x_3, 3x_1 + x_2, 2x_2 + x_3)$$ for every $$(x_1,x_2,x_3) \in \mathbb{R^3}$$

I am confused. I do not know how to start the problem. I have researched the description but I cannot find anything definitive for this type of problem.

Update:

$$ A = \left[\begin{array}{rrr} 1 & 1 & 1\\ 0 & 1 & 1\\ 3 & 1 & 0\\ 0 & 2 & 1\\ \end{array}\right] $$

$$ ref(A) = \left[\begin{array}{rrr} 1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1\\ 0 & 0 & 0\\ \end{array}\right] $$

From $T(e_1)=(1,0,3,0)^T, T(e_2)=(1,1,1,2)^T, T(e_3)=(1,1,0,1)^T$

Is this correct? Or am I completely lost?

How would I find the dimension of the kernel of $T$ and the range of $T$?

Answer 1

We are asked to find the standard matrix $A$ for $T$:

Consider the transformation $T : \mathbb{R^3} \rightarrow \mathbb{R^4}$ given by

$$T(x_1, x_2, x_3) = (x_1 + x_2 + x_3, x_2 + x_3, 3x_1 + x_2, 2x_2 + x_3)$$ for every $$(x_1,x_2,x_3) \in \mathbb{R^3}$$

Your update is correct $A=\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 3 & 1 & 0 \\ 0 & 2 & 1 \end{bmatrix}$.

We are basically solving for $T(x)=Ax$ where $x$ is the vector containing entries $(x_{1},x_{2},x_{3})$ and $A$ is the standard matrix for $T$.

$$\begin{align}T(x) &= T(x_1, x_2, x_3) \\ &= (x_1 + x_2 + x_3, x_2 + x_3, 3x_1 + x_2, 2x_2 + x_3)\\ &=\begin{bmatrix}x_{1}+x_{2}+x_{3} \\ 0x_{1}+x_{2}+x_{3} \\3x_{1} +x_{2} +0x_{3} \\ 0x_{1}+2x_{2}+x_{3}\end{bmatrix}\\ &=x_{1}\begin{bmatrix} 1\\0\\3\\0\end{bmatrix}+x_{2}\begin{bmatrix}1\\1\\1\\2\end{bmatrix} +x_{3}\begin{bmatrix}1\\1\\0\\1\end{bmatrix}\\ &=\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 3 & 1 & 0 \\ 0 & 2 & 1 \end{bmatrix}\begin{bmatrix} x_{1}\\x_{2}\\x_{3}\end{bmatrix} \\ &=Ax \\ \end{align}$$

Thus, $A=\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 3 & 1 & 0 \\ 0 & 2 & 1 \end{bmatrix}$.