The following summarizes and clarifies info already presented or referenced in this thread.
Define three vectors {V1, V2, V3 ∈ \mathbb{R}^{3}
} by the following:
(Note: This definition differs from OP's.) xanthym did tell you the notation had changed.
1: \ \ \ \ \vec{V}_{1} \ = \ \left [ \begin{array}{ccc}a_{1} \\b_{1} \\c_{1} \\\end{array}\right ]\ \ \ \ \ \ \vec{V}_{2} \ = \ \left [ \begin{array}{ccc}a_{2} \\b_{2} \\c_{2} \\\end{array}\right ]\ \ \ \ \ \ \vec{V}_{3} \ = \ \left [ \begin{array}{ccc}a_{3} \\b_{3} \\c_{3} \\\end{array}\right ]
These are xanthym's vectors.
1: \ \ \ \ \vec{V}_{1} \ = \ \left [ \begin{array}{ccc}a_{1} \\a_{2} \\a_{3} \\\end{array}\right ]\ \ \ \ \ \ \vec{V}_{2} \ = \ \left [ \begin{array}{ccc}b_{1} \\b_{2} \\b_{3} \\\end{array}\right ]\ \ \ \ \ \ \vec{V}_{3} \ = \ \left [ \begin{array}{ccc}c_{1} \\c_{2} \\c_{3} \\\end{array}\right ]
These are your vectors.
These vectors are Linearly Independent if and only if the scalar triple (x, y, z)=(0, 0, 0) is the SINGLE UNIQUE solution to:
2: \ \ \ \ x\vec{V}_{1} \ + \ y\vec{V}_{2} \ + \ z\vec{V}_{3} \, \ = \ \, \vec{\mathbf{0}}
This is xanthym's linear combination
2: \ \ \ \ A\vec{V}_{1} \ + \ B\vec{V}_{2} \ + \ C\vec{V}_{3} \, \ = \ \, \vec{\mathbf{0}}
This is the corresponding linear combination of your vectors. As a matter of personal preference, I would never use x, y, and z as scalar coefficients, so I have used A, B, and C. This is not a matter of right and wrong. It is a preference to avoid potential confusion between scalar coefficients, and the functions x, y, and z
Or equivalently, the single unique solution to:
3: \ \ \ \ \ x\left [ \begin{array}{ccc}a_{1} \\b_{1} \\c_{1} \\\end{array}\right ]\ + \ y\left [ \begin{array}{ccc}a_{2} \\b_{2} \\c_{2} \\\end{array}\right ]\ + \ z\left [ \begin{array}{ccc}a_{3} \\b_{3} \\c_{3} \\\end{array}\right ] \ \ = \ \ \left [ \begin{array}{ccc}0 \\0 \\0 \\\end{array}\right ]
These are xanthym's equations.
3: \ \ \ \ \ A\left [ \begin{array}{ccc}a_{1} \\a_{2} \\a_{3} \\\end{array}\right ]\ + \ B\left [ \begin{array}{ccc}b_{1} \\b_{2} \\b_{3} \\\end{array}\right ]\ + \ C\left [ \begin{array}{ccc}c_{1} \\c_{2} \\c_{3} \\\end{array}\right ] \ \ = \ \ \left [ \begin{array}{ccc}0 \\0 \\0 \\\end{array}\right ]
These are the equivalent equations for your vectors.
Or equivalently, that (x, y, z)=(0, 0, 0) is the single unique solution to:
4: \ \ \ \ \begin{array}{ccccccc}a_1 x & + & a_2 y & + & a_3 z & \, = \, & 0 \\b_1 x & + & b_2 y & + & b_3 z & \, = \, & 0 \\c_1 x & + & c_2 y & + & c_3 z & \, = \, & 0 \\
These are xanthym's equations.
4: \ \ \ \ \begin{array}{ccccccc}a_1 A & + & b_1 B & + & c_1 C & \, = \, & 0 \\a_2 A & + & b_2 B & + & c_2 C & \, = \, & 0 \\a_3 A & + & b_3 B & + & c_3 C & \, = \, & 0 \\
These are the equivalent equations for your vectors. Now, if you replaced the A, B, and C that I prefer with the x, y, and z that xanthym used, you can see that your equations are NOT the same equations. The only possible conclusion based on your original equations is that you were using x, y, and z as a basis for representing your three vectors. This is completely different from using x, y, and z as scalar coefficients the way xanthym used them.
The triple (x, y, z) will be the single unique solution to Eq #4 if and only if the determinant of its coefficients is NON-zero:
5: \color{red}\ \ \ \ \ \ \left | \begin{array}{ccc}a_{1} & a_{2} & a_{3} \\b_{1} & b_{2} & b_{3} \\c_{1} & c_{2} & c_{3} \\\end{array}\right | \ \ \neq \ \ \textsf{0} \ \ \ \color{black} \Longleftrightarrow \ \ \ \color{red} \{\vec{V}_{1}, \ \vec{V}_{2}, \ \vec{V}_{3} \ \ \ \textsf{Linearly Independent} \}
These are xanthym's statement and equations.
The triple (A, B, C) will be the single unique solution to Eq #4 if and only if the determinant of its coefficients is NON-zero:
5: \color{red}\ \ \ \ \ \ \left | \begin{array}{ccc}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3} \\\end{array}\right | \ \ \neq \ \ \textsf{0} \ \ \ \color{black} \Longleftrightarrow \ \ \ \color{red} \{\vec{V}_{1}, \ \vec{V}_{2}, \ \vec{V}_{3} \ \ \ \textsf{Linearly Independent} \}
These are the equivalent statement and equations for your vectors.
Eq #5 probably provides the easiest method for determining Linear Independence in .
For convenience, Left Side of Eq #5 is rapidly determined from:
6: \ \ \ \ \left [ \begin{array}{ccc}a_{1} & a_{2} & a_{3} \\b_{1} & b_{2} & b_{3} \\c_{1} & c_{2} & c_{3} \\\end{array}\right ] \ \ = \ \ \left [ \ \vec{V}_{1} \ \ \vec{V}_{2} \ \ \vec{V}_{3} \ \right ]
This is xanthym's equation.
6: \ \ \ \ \left [ \begin{array}{ccc}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3} \\\end{array}\right ] \ \ = \ \ \left [ \ \vec{V}_{1} \ \ \vec{V}_{2} \ \ \vec{V}_{3} \ \right ]
This is your equivalent equation.
Note that only 3 possibilities exist for Eq #4 solutions:
1) Single, Unique Solution
2) Infinite # of Solutions
3) No Solutions
Eq #5 indicates situation #1.
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