1 point Which of the following sets of vectors are linearly independent Check the boxes for linearly independent sets A ft ft c 6 10 ft l 3 5 B ft ft l9 2 ft c 7 4 ft l3 5 c ft ft c1 9 ft c 2 1 D ft...

#### Solution By Steps ***Step 1: Define Linear Independence*** A set of vectors is linearly independent if the only way to form the zero vector as a linear combination of these vectors is by setting all coefficients to zero. ***Step 2: Check Set A*** For set A: $\left\{\left[\begin{array}{c}-6 \\ -10\end{array}\right],\left[\begin{array}{l}-3 \\ -5\end{array}\right]\right\}$ To check linear independence, set up the equation $c_1\left[\begin{array}{c}-6 \\ -10\end{array}\right] + c_2\left[\begin{array}{l}-3 \\ -5\end{array}\right] = \left[\begin{array}{c}0 \\ 0\end{array}\right]$ and solve for $c_1$ and $c_2$. ***Step 3: Check Set B*** For set B: $\left\{\left[\begin{array}{l}9 \\ 2\end{array}\right],\left[\begin{array}{c}-7 \\ 4\end{array}\right],\left[\begin{array}{l}3 \\ 5\end{array}\right]\right\}$ Set up the equation $c_1\left[\begin{array}{l}9 \\ 2\end{array}\right] + c_2\left[\begin{array}{c}-7 \\ 4\end{array}\right] + c_3\left[\begin{array}{l}3 \\ 5\end{array}\right] = \left[\begin{array}{c}0 \\ 0\end{array}\right]$ and solve for $c_1$, $c_2$, and $c_3$. ***Step 4: Check Set C*** For set C: $\left\{\left[\begin{array}{c}1 \\ -9\end{array}\right],\left[\begin{array}{c}-2 \\ 1\end{array}\right]\right\}$ Set up the equation $c_1\left[\begin{array}{c}1 \\ -9\end{array}\right] + c_2\left[\begin{array}{c}-2 \\ 1\end{array}\right] = \left[\begin{array}{c}0 \\ 0\end{array}\right]$ and solve for $c_1$ and $c_2$. ***Step 5: Check Set D*** For set D: $\left\{\left[\begin{array}{c}7 \\ -3 \\ 9\end{array}\right],\left[\begin{array}{c}2 \\ -7 \\ 4\end{array}\right],\left[\begin{array}{l}-5 \\ -4 \\ -5\end{array}\right]\right\}$ Set up the equation $c_1\left[\begin{array}{c}7 \\ -3 \\ 9\end{array}\right] + c_2\left[\begin{array}{c}2 \\ -7 \\ 4\end{array}\right] + c_3\left[\begin{array}{l}-5 \\ -4 \\ -5\end{array}\right] = \left[\begin{array}{c}0 \\ 0 \\ 0\end{array}\right]$ and solve for $c_1$, $c_2$, and $c_3$. ***Step 6: Check Set E*** For set E: $\left\{\left[\begin{array}{l}0 \\ 0\end{array}\right],\left[\begin{array}{c}7 \\ -4\end{array}\right]\right\}$ Set up the equation $c_1\left[\begin{array}{l}0 \\ 0\end{array}\right] + c_2\left[\begin{array}{c}7 \\ -4\end{array}\right] = \left[\begin{array}{c}0 \\ 0\end{array}\right]$ and solve for $c_1$ and $c_2$. ***Step 7: Check Set F*** For set F: $\left\{\left[\begin{array}{l}5 \\ 2 \\ 4\end{array}\right],\left[\begin{array}{l}0 \\ 7 \\ 6\end{array}\right],\left[\begin{array}{c}-1 \\ 8 \\ -3\end{array}\right]\right\}$ Set up the equation $c_1\left[\begin{array}{l}5 \\ 2 \\ 4\end{array}\right] + c_2\left[\begin{array}{l}0 \\ 7 \\ 6\end{array}\right] + c_3\left[\begin{array}{c}-1 \\ 8 \\ -3\end{array}\right] = \left[\begin{array}{c}0 \\ 0 \\ 0\end{array}\right]$ and solve for $c_1$, $c_2$, and $c_3. #### Final Answer The linearly independent sets are: A. $\left\{\left[\begin{array}{c}-6 \\ -10\end{array}\right],\left[\begin{array}{l}-3 \\ -5\end{array}\right]\right\}$ C. $\left\{\left[\begin{array}{c}1 \\ -9\end{array}\right],\left[\begin{array}{c}-2 \\ 1\end{array}\right]\right\}$ E. $\left\{\left[\begin{array}{l}0 \\ 0\end{array}\right],\left[\begin{array}{c}7 \\ -4\end{array}\right]\right\}$ #### Key Concept Linear Independence #### Key Concept Explanation Linear independence of vectors means that no vector in the set can be represented as a linear combination of the others. This property is crucial in various mathematical and engineering applications, ensuring unique solutions and efficient computations in systems of equations and transformations.