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
To express v = (3, 7, -4) as a linear combination of v_1 = (1, 2, 3), v_2 = (2, 3, 7), and v_3 = (3, 5, 6), we need to find scalars a_1, a_2, a_3 such that: v = a_1 v_1 + a_2 v_2 + a_3 v_3 (3, 7, -4) = a_1 (1, 2, 3) + a_2 (2, 3, 7) + a_3 (3, 5, 6) Step 1: Formulate the system of linear equations. Expanding the vector equation gives: (3, 7, -4) = (a_1 + 2a_2 + 3a_3, 2a_1 + 3a_2 + 5a_3, 3a_1 + 7a_2 + 6a_3) Equating the components, we get the following system: a_1 + 2a_2 + 3a_3 &= 3 \\ 2a_1 + 3a_2 + 5a_3 &= 7 \\ 3a_1 + 7a_2 + 6a_3 &= -4 Step 2: Write the augmented matrix for the system. 1 & 2 & 3 & | & 3 \\ 2 & 3 & 5 & | & 7 \\ 3 & 7 & 6 & | & -4 Step 3: Perform row operations to achieve row echelon form. Apply R_2 R_2 - 2R_1 and R_3 R_3 - 3R_1: 1 & 2 & 3 & | & 3 \\ 0 & -1 & -1 & | & 1 \\ 0 & 1 & -3 & | & -13 Apply R_3 R_3 + R_2: 1 & 2 & 3 & | & 3 \\ 0 & -1 & -1 & | & 1 \\ 0 & 0 & -4 & | & -12 Step 4: Solve the system using back-substitution. From the third row: -4a_3 = -12 a_3 = (-12)/(-4) a_3 = 3 From the second row: -a_2 - a_3 = 1 -a_2 - 3 = 1 -a_2 = 4 a_2 = -4 From the first row: a_1 + 2a_2 + 3a_3 = 3 a_1 + 2(-4) + 3(3) = 3 a_1 - 8 + 9 = 3 a_1 + 1 = 3 a_1 = 2 The scalars are a_1 = 2, a_2 = -4, and a_3 = 3. The final answer is a_1 = 2, a_2 = -4, a_3 = 3. Send me the next one 📸