<s>[INST] Finding the Standard Matrix for Linear Transformation 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 $T$?
 [/INST] 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, $A=\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 3 & 1 & 0 \\ 0 & 2 & 1 \end{bmatrix}$.
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