Find the image of the vector under the linear transformation with the matrix
Find the image of the vector $v=[1,2,2,1]$ under the linear transformation with the matrix $\begin{bmatrix}1 & 0 & 2 & 1\\2 & 1 & 0 & 0\\3 & 1 & 1 & 2\\1 & 2 & 1 & 3\end{bmatrix}$
Explanations would be nice, but I would also like the answer and the steps to get to the answer, so that I know how to do the problem.
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$\begingroup$The linear transformation defined by a matrix $A$ is the function on a (column-)vector $x$ defined by $$ L_A(\vec x) =A \vec x = \pmatrix{a_{11} & \cdots & a_{1n}\\\vdots&\ddots & \vdots\\ a_{n1} & \cdots & a_{nn}} \;\pmatrix{x_1\\ \vdots\\x_n} = \pmatrix{a_{11}x_1 + \cdots + a_{1n}x_n\\ \vdots \\ a_{n1} x_1 + \cdots + a_{nn}x_n} $$ So, what they are asking you to do is find the product $$ L_A(\vec v) = A \vec v = \pmatrix{ 1 & 0 & 2 & 1\\2 & 1 & 0 & 0\\3 & 1 & 1 & 2\\1 & 2 & 1 & 3 }\; \pmatrix{1\\2\\2\\1} $$
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