Answer to Question #167940 in Linear Algebra for Zeeshan

Question #167940

Express v=(2,-5,3) in R³ as a linear combination of the vectors U₁ =(1,-3,2) , U₂=(2,-4,-1), U₃=(1,-5,7)

Expert's answer

Rewriting the equation in the form,

"\\begin{bmatrix}\n 2 \\\\\n -5\\\\\n 3\n\\end{bmatrix}= x\\begin{bmatrix}\n 1 \\\\\n -3\\\\\n 2\n\\end{bmatrix}+y\\begin{bmatrix}\n 2 \\\\\n -4\\\\\n -1\n\\end{bmatrix}+z\\begin{bmatrix}\n 1 \\\\\n -5\\\\\n 7\n\\end{bmatrix}\\\\\n\\begin{bmatrix}\n x + 2y + z \\\\\n -3x -4y -5z \\\\\n 2x -y +7z\n\\end{bmatrix}=\\begin{bmatrix}\n 2 \\\\\n -5\\\\\n 3\n\\end{bmatrix}\\\\\n\\text{reducing the matrix from the back yields}\\\\.\n\\begin{bmatrix}\n x + 2y + z=2 \\\\\n -3x -4y -5z=-5 \\\\\n 2x -y +7z=3\n\\end{bmatrix}\\\\\n-2R_1+R_3 \\implies R_3\\\\\n3R_1+R_2 \\implies R_2\\\\\n\\begin{bmatrix}\n x + 2y + z=2 \\\\\n 2y -2z=1 \\\\\n -5y +5z=-1\n\\end{bmatrix}\\\\\n\\frac{5}{2}R_2+R_3 \\implies R_3\\\\\n\\\\\n\\begin{bmatrix}\n x + 2y + z=2 \\\\\n 2y -2z=1 \\\\\n 0=\\frac{3}{2}\n\\end{bmatrix}\\\\\n\\text{The system has no solution.}\\\\\n\\text{Thus it is impossible to write the vector v as a linear combination of}\\\\u_1, u_2, u_3"