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
let's determine the linear dependence of these vectors. 3. Determine whether or not the following vectors in R^3 are linearly dependent: u = (1,2,5), v = (2,5,1), and w = (1,5,2). To determine if the vectors are linearly dependent, we can form a matrix with these vectors as columns and calculate its determinant. If the determinant is zero, the vectors are linearly dependent. If it's non-zero, they are linearly independent. Step 1: Form the matrix A with the given vectors as columns. A = 1 & 2 & 1 \\ 2 & 5 & 5 \\ 5 & 1 & 2 Step 2: Calculate the determinant of matrix A. We will use cofactor expansion along the first row: (A) = 1 · 5 & 5 \\ 1 & 2 - 2 · 2 & 5 \\ 5 & 2 + 1 · 2 & 5 \\ 5 & 1 Step 3: Compute the 2 × 2 determinants. 5 & 5 \\ 1 & 2 = (5 · 2) - (5 · 1) = 10 - 5 = 5 2 & 5 \\ 5 & 2 = (2 · 2) - (5 · 5) = 4 - 25 = -21 2 & 5 \\ 5 & 1 = (2 · 1) - (5 · 5) = 2 - 25 = -23 Step 4: Substitute these values back into the determinant formula and simplify. (A) = 1 · (5) - 2 · (-21) + 1 · (-23) (A) = 5 + 42 - 23 (A) = 47 - 23 (A) = 24 Step 5: Conclude based on the determinant value. Since (A) = 24 ≠ 0, the vectors are linearly independent. The vectors are linearly independent What's next?