Answer to Question #106164 in Analytic Geometry for Jacky

Question #106164
v1(3,-2,-2), v2(2,-1,-2), v3(-1,1,1) , u(0,1,h)
find all values of h such that u is a linear combination of v1, v2 and v3
1
Expert's answer
2020-03-23T14:04:30-0400

First we check if system {v1,v2,v3}\{v_1, v_2, v_3 \} is linearly independent.

(322212111)=R1R2+R3(001212111)=R2+2R3(001010111)=R3R3,R3+R1+R2=(001010100)\begin{pmatrix} 3 & -2 & -2 \\ 2 & -1 & -2 \\ -1 & 1 & 1 \end{pmatrix} \overset{R_1 - R_2 + R_3}{=} \begin{pmatrix} 0 & 0 & 1 \\ 2 & -1 & -2 \\ -1 & 1 & 1 \end{pmatrix} \overset{R_2 + 2R_3}{=} \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 1 & 1 \end{pmatrix} \overset{R_3 \to -R_3, R_3 + R_1 + R_2}{=}\\ =\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}

Thus, space of linear combinations of v1,v2,v3v_1, v_2, v_3 includes all vectors of R3R^3, so h can take on any value.


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