Answer:
The value of the constant [tex]k[/tex] so that [tex]\vec u_{3}[/tex] is a linear combination of [tex]\vec u_{1}[/tex] and [tex]\vec u_{2}[/tex] is [tex]\frac{7}{10}[/tex].
Step-by-step explanation:
Let be [tex]\vec u_{1} = [2,3,1][/tex], [tex]\vec u_{2} = [4,1,0][/tex] and [tex]\vec u_{3} = [1, 2,k][/tex], [tex]\vec u_{3}[/tex] is a linear combination of [tex]\vec u_{1}[/tex] and [tex]\vec u_{3}[/tex] if and only if:
[tex]\alpha_{1} \cdot \vec u_{1} + \alpha_{2} \cdot \vec u_{2} +\alpha_{3}\cdot \vec u_{3} = \vec O[/tex] (Eq. 1)
Where:
[tex]\alpha_{1}[/tex], [tex]\alpha_{2}[/tex], [tex]\alpha_{3}[/tex] - Scalar coefficients of linear combination, dimensionless.
By dividing each term by [tex]\alpha_{3}[/tex]:
[tex]\lambda_{1}\cdot \vec u_{1} + \lambda_{2}\cdot \vec u_{3} = -\vec u_{3}[/tex]
[tex]\vec u_{3}=-\lambda_{1}\cdot \vec u_{1}-\lambda_{2}\cdot \vec u_{2}[/tex] (Eq. 2)
[tex]\vec O[/tex] - Zero vector, dimensionless.
And all vectors are linearly independent, meaning that at least one coefficient must be different from zero. Now we expand (Eq. 2) by direct substitution and simplify the resulting expression:
[tex][1,2,k] = -\lambda_{1}\cdot [2,3,1]-\lambda_{2}\cdot [4,1,0][/tex]
[tex][1,2,k] = [-2\cdot\lambda_{1},-3\cdot \lambda_{1},-\lambda_{1}]+[-4\cdot \lambda_{2},-\lambda_{2},0][/tex]
[tex][0,0,0] = [-2\cdot \lambda_{1},-3\cdot \lambda_{1},-\lambda_{1}]+[-4\cdot \lambda_{2},-\lambda_{2},0]+[-1,-2,-k][/tex]
[tex][-2\cdot \lambda_{1}-4\cdot \lambda_{2}-1,-3\cdot \lambda_{1}-\lambda_{2}-2,-\lambda_{1}-k] =[0,0,0][/tex]
The following system of linear equations is obtained:
[tex]-2\cdot \lambda_{1}-4\cdot \lambda_{2}= 1[/tex] (Eq. 3)
[tex]-3\cdot \lambda_{1}-\lambda_{2}= 2[/tex] (Eq. 4)
[tex]-\lambda_{1}-k = 0[/tex] (Eq. 5)
The solution of this system is:
[tex]\lambda_{1} = -\frac{7}{10}[/tex], [tex]\lambda_{2} = \frac{1}{10}[/tex], [tex]k = \frac{7}{10}[/tex]
The value of the constant [tex]k[/tex] so that [tex]\vec u_{3}[/tex] is a linear combination of [tex]\vec u_{1}[/tex] and [tex]\vec u_{2}[/tex] is [tex]\frac{7}{10}[/tex].
