Question #260439

Write v = (2, -5 , 3) as a linear combination of

u1=(1,−3 ,2)

u2=(2,−4 ,−1)

u3=(1,−3 , 7)


1
Expert's answer
2021-11-04T20:15:34-0400

Let x1u1+x2u2+x3u3=vxRx_1u_1+x_2u_2+x_3u_3=v\quad x\in\reals


[121343217][x1x2x3]=[253]\begin{bmatrix} 1&2&1 \\ -3&-4&-3\\2&-1&7 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2\\x_3 \end{bmatrix}=\begin{bmatrix} 2 \\-5\\3 \end{bmatrix}


Considering augmented matrix AA for this system and applying Gauss- Jordan elimination


A=121343217253A=\begin{matrix} |&1&2&1\\|&-3&-4&-3&\\|&2&-1&7 \end{matrix}\begin{vmatrix} 2\\-5\\3 \end{vmatrix}


13-\frac{1}{3} R2R1R2R_2-R_1\to\>R_2


12102301127221332\begin{matrix} |&1&2&1 \\ |&0&-\frac{2}{3} & 0\\ |&1&-\frac{1}{2}&\frac{7}{2} \end{matrix}\begin{vmatrix} 2\\ -\frac{1}{3}\\ \frac{3}{2} \end{vmatrix}



12R3R1R2\frac{1}{2}\>R_3-R_1\>\to\>R_2



12102300525221312\begin{matrix} |&1&2&1 \\ |&0&-\frac{2}{3}&0 \\ |&0&-\frac{5}{2}&\frac{5}{2} \end{matrix}\begin{vmatrix} 2 \\ -\frac{1}{3} \\ -\frac{1}{2} \end{vmatrix}




32R2R2-\frac{3}{2}\>R_2\to\>R_2 1210100525221212\begin{matrix} |&1&2&1 \\ |&0&1&0 \\ |&0&-\frac{5}{2}&\frac{5}{2} \end{matrix}\begin{vmatrix} 2 \\ \frac{1}{2} \\ -\frac{1}{2} \end{vmatrix}


25R3R2R3-\frac{2}{5}R_3-R_2\to\>R_3


121010001212310\begin{matrix} |&1&2&1 \\ |&0&1&0 \\ |&0&0&-1 \end{matrix}\begin{vmatrix} 2 \\ \frac{1}{2} \\ -\frac{3}{10} \end{vmatrix}



1-1 R3R3R_3\to\>R_3 121010001212310\begin{matrix} |&1&2&1 \\ |&0&1&0 \\ |&0&0&1 \end{matrix}\begin{vmatrix} 2 \\ \frac{1}{2}\\ \frac{3}{10} \end{vmatrix}



R1R3R1R_1-R_3\to\>R_1 120010001212310\begin{matrix} |&1&2&0 \\ |&0&1&0\\ |&0&0&1 \end{matrix}\begin{vmatrix} 2 \\ \frac{1}{2} \\ \frac{3}{10} \end{vmatrix}



R12R2R1R_1-2R_2\to\>R_1 10001000171012310\begin{matrix} |&1&0&0 \\ |&0&1&0 \\ |&0&0&1 \end{matrix}\begin{vmatrix} \frac{7}{10} \\ \frac{1}{2} \\ \frac{3}{10} \end{vmatrix}


rref A=A= 10001000171012310\begin{matrix}|& 1&0&0 \\|& 0&1&0\\|&0&0&1 \end{matrix}\begin{vmatrix} \frac{7}{10}\\\frac{1}{2}\\\frac{3}{10} \end{vmatrix}


    x1=710x2=12x3=310\implies x_1=\frac{7}{10}\quad x_2=\frac{1}{2} \quad x_3=\frac{3}{10}



v=710u1+12u2+310u3v=\frac{7}{10}u_1+\frac{1}{2}u_2+\frac{3}{10}u_3


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