This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Determine linear independence for 4.1.2. To determine if the vectors x_1 = 1 \\ 2 \\ 3 , x_2 = 3 \\ 2 \\ 9 , and x_3 = 5 \\ 2 \\ -1 are linearly independent, we can form a matrix M with these vectors as columns and calculate its determinant. If the determinant is non-zero, the vectors are linearly independent. M = 1 & 3 & 5 \\ 2 & 2 & 2 \\ 3 & 9 & -1 Calculate the determinant of M: (M) = 1 · 2 & 2 \\ 9 & -1 - 3 · 2 & 2 \\ 3 & -1 + 5 · 2 & 2 \\ 3 & 9 (M) = 1 · ((2)(-1) - (2)(9)) - 3 · ((2)(-1) - (2)(3)) + 5 · ((2)(9) - (2)(3)) (M) = 1 · (-2 - 18) - 3 · (-2 - 6) + 5 · (18 - 6) (M) = 1 · (-20) - 3 · (-8) + 5 · (12) (M) = -20 + 24 + 60 (M) = 64 Since (M) = 64 ≠ 0, the vectors are linearly independent. The set of vectors is linearly independent. Step 2: Determine linear independence for 4.1.3. To determine if the vectors x_1 = 4 \\ -1 \\ 2 , x_2 = 3 \\ 2 \\ 3 , and x_3 = 5 \\ 2 \\ 4 are linearly independent, we form a matrix M with these vectors as columns and calculate its determinant. M = 4 & 3 & 5 \\ -1 & 2 & 2 \\ 2 & 3 & 4 Calculate the determinant of M: (M) = 4 · 2 & 2 \\ 3 & 4 - 3 · -1 & 2 \\ 2 & 4 + 5 · -1 & 2 \\ 2 & 3 (M) = 4 · ((2)(4) - (2)(3)) - 3 · ((-1)(4) - (2)(2)) + 5 · ((-1)(3) - (2)(2)) (M) = 4 · (8 - 6) - 3 · (-4 - 4) + 5 · (-3 - 4) (M) = 4 · (2) - 3 · (-8) + 5 · (-7) (M) = 8 + 24 - 35 (M) = 32 - 35 (M) = -3 Since (M) = -3 ≠ 0, the vectors are linearly independent. The set of vectors is linearly independent. 3 done, 2 left today. You're making progress.