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Let \(f\) be the transformation of \(\mathbb{R}^{2}\) given…

Let \(f\) be the transformation of \(\mathbb{R}^{2}\) given by rotating \(50^{\circ}\) counterclockwise around the origin. Find the standard matrix for \(f\) and then use that to find where the point \(\left(1,4\right)\) gets sent under this rotation. (As \(50^{\circ}\) is not a special angle, you will have to use decimal approximations.)

  Exam 2 Definitions 1.) [5 points each] Complete both of th…

  Exam 2 Definitions 1.) [5 points each] Complete both of the following definitions. You ONLY have to write out what correctly finishes the definition, not the part that is given. a.) If \(S=\left\{v_{1},\ldots ,v_{k}\right\}\) is a set of vectors in a vector space \(V\), then \(S\) is linearly independent if: b.) If \(S=\left\{v_{1},\ldots ,v_{k}\right\}\) is a set of vectors in a vector space \(V\), then \(S\) is a basis for \(V\) if: The use of math tags around this note triggers MathJax to display the LaTex written on this page as MathML objects.

Let \(V=\mathbb{R}^{2}\) with the following operations \[\be…

Let \(V=\mathbb{R}^{2}\) with the following operations \[\begin{align} \left(x_{1},y_{1}\right)\oplus\left(x_{2},y_{2}\right)&=\left(x_{1}+x_{2}+1,y_{1}+y_{2}-1\right)\\k\otimes\left(x,y\right)&=\left(kx,ky\right) \end{align}\] Show that \(V\) is not a vector space by showing that it does not satisfy \[ k\otimes\left(\overrightarrow{a}\oplus\overrightarrow{b}\right) =\left(k\otimes\overrightarrow{a}\right)\oplus\left(k\otimes\overrightarrow{b}\right)\]