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📰 Solution: The matrix $\mathbf{M}$ is constructed by placing the images of the standard basis vectors as its columns. Thus, $\mathbf{M} = \begin{pmatrix} 2 & 3 \\ -1 & 4 \end{pmatrix}$. Verifying, $\mathbf{M} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$ and $\mathbf{M} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$, confirming correctness. $\boxed{\begin{pmatrix} 2 & 3 \\ -1 & 4 \end{pmatrix}}$ 📰 Question: An environmental consultant models a river's flow as the line $y = -\frac{1}{2}x + 5$. Find the point on this line closest to the pollution source at $(4, 3)$. 📰 Solution: The closest point is the projection of $(4, 3)$ onto the line. The formula for the projection of a point $(x_0, y_0)$ onto $ax + by + c = 0$ is used. Rewriting the line as $\frac{1}{2}x + y - 5 = 0$, we compute the projection. Alternatively, parametrize the line and minimize distance. Let $x = t$, then $y = -\frac{1}{2}t + 5$. The squared distance to $(4, 3)$ is $(t - 4)^2 + \left(-\frac{1}{2}t + 5 - 3\right)^2 = (t - 4)^2 + \left(-\frac{1}{2}t + 2\right)^2$. Expanding: $t^2 - 8t + 16 + \frac{1}{4}t^2 - 2t + 4 = \frac{5}{4}t^2 - 10t + 20$. Taking derivative and setting to zero: $\frac{5}{2}t - 10 = 0 \Rightarrow t = 4$. Substituting back, $y = -\frac{1}{2}(4) + 5 = 3$. Thus, the closest point is $(4, 3)$, which lies on the line. $\boxed{(4, 3)}$ 📰 The Surprising Secret To Making Jello Thats Bold Color Filled And Unforgettable 📰 The Surprising Story Behind Hold On Till May The Lyrics You Didnt Know Were Revolutionary 📰 The Surprising Truth About How Long A School Bus Is Dont Miss This Eye Opening Fact 📰 The Surprising Truth About Kratos Height880 Inches You Wont Believe It 📰 The Surprising Truth Behind Horse With No Name Chordschords That Sound Like A Horses Roar 📰 The Surprising Truth How Far Is A Klick Discover The Mind Blowing Answer Now 📰 The Surprising Truth Milligrams In A Teaspooncatch This Essential Fact 📰 The Surprising Truth Was Jesus A Giant Scientists Reveal The Hidden Height Secrets 📰 The Surprising Way To Measure A Tv Trusted By Experts 📰 The Temperature In A City Drops From 75F To 55F Over 4 Hours What Is The Average Rate Of Temperature Decrease Per Hour 📰 The Terrible Truth About How Long Playstation Network Will Be Offdont Miss This 📰 The Terrifying Truth The Complete Count Of Friday The 13Th Movies Youve Never Seen Before 📰 The Time Required Is 480 Liters 8 Liters Per Minute 60 Minutes 📰 The Total Temperature Drop Is 75F 55F 20F 📰 The Tricky Way To Write An Address On An Envelope You Need To See This