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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. $\boxed{(4, 3)}$ 📰 Wesley Snipe Stuns Us Again Here Are His Hottest New Films You Need To Watch Now 1161326 📰 Given The Confusion Lets Assume The Function Is Meant To Be Minimized As Total Cost But The Problem Says Cost Per Unit 529726 📰 Youll Never Guess How Simple The Afo Brace Solves Daily Chaos 5814348 📰 You Wont Believe How Addictive These Free Classic Mahjong Crazy Games Are Play Now 842935 📰 Home Depot Pro 8000757 📰 Java Se Specification Secrets You Must Know Before Coding Like A Pro 2762531 📰 Curaleaf Ticker 6093362 📰 How Old Is Chris Brown 4955190 📰 Diabetes Breakfast Meal 9972212 📰 You Wont Believe What Happens When You Try The Netflix Bet Netflicks Secrets Exposed 6679155 📰 Wesbanco Stock Price Jumps 50Heres Why This Breakthrough Will Change Market Dynamics 2594873 📰 Brown And Light Brown The Underrated Combination Thats Taking Interior Trends By Storm 4682470 📰 5Laptop Mouse Touchpad Not Working These Simple Fixes Restore Functionality Today 8654884 📰 Servicio Al Cliente De Wells Fargo 6115682 📰 5 New Free Jackpot Party Slots Slide In Winning Coins Live The Party 778486