If $S = \{u_1, u_2\}$ is linearly independent, then the following conditions hold:

Conditions for Linear Independence:

1. Definition: Vectors $u_1$ and $u_2$ are linearly independent if the only scalars $a$ and $b$ such that:

   $a u_1 + b u_2 = 0$

   are $a = 0$ and $b = 0$.

2. Geometric Interpretation: The vectors $u_1$ and $u_2$ are not scalar multiples of each other. This implies that $u_1$ and $u_2$ span a two-dimensional space, assuming the underlying vector space has dimension at least 2.

3. Determinants (for 2D or 3D vectors): If $u_1$ and $u_2$ are given in component form, say $u_1 = (x_1, y_1)$ and $u_2 = (x_2, y_2)$ in 2D, then:

   $\text{Determinant: } \det\begin{bmatrix}x_1 & x_2 \\ y_1 & y_2\end{bmatrix} \neq 0.$

   This determinant being non-zero confirms that $u_1$ and $u_2$ are linearly independent.

4. Rank (for matrices): If $u_1$ and $u_2$ are rows or columns of a matrix, the matrix must have full rank (rank = 2 for a 2D space).

5. Projection: The projection of $u_1$ onto $u_2$ or vice versa is not equal to the full vector, meaning no vector lies completely along the other.

Summary:

The set $S = \{u_1, u_2\}$ is linearly independent if no non-trivial linear combination of $u_1$ and $u_2$ results in the zero vector, and neither vector is a scalar multiple of the other.

Would you like me to expand on conditions, examples, or geometric interpretations?

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Related Questions:

1. What are specific examples of linearly independent vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$?
2. How is the concept of linear independence applied in solving systems of linear equations?
3. Can we test linear independence for more than two vectors? If so, how?
4. What role does the determinant play in confirming linear independence for a set of vectors?
5. How does the Gram-Schmidt process help in constructing a linearly independent set?

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Tip:

Always remember that a set of vectors is linearly independent if none of the vectors can be expressed as a linear combination of the others. This ensures they form a basis for the subspace they span.