91Ó°ÊÓ

The idea of a linear combination involves taking two or more vectors and combining them using scalar multiplication and addition. Imagine these vectors as arrows pointing in space. By scaling these arrows using numbers (scalars) and then adding them together, you can create new arrows or vectors. For the vectors \( \mathbf{v}_{1} \) and \( \mathbf{v}_{2} \) given in the exercise, a linear combination looks like \( a\mathbf{v}_{1} + b\mathbf{v}_{2} \). Here, \( a \) and \( b \) are scalars that we choose.

• When \( a = 0 \) and \( b = 0 \), we get the zero vector, a vector with no length.
• If we let \( a \) or \( b \) increase or decrease, we stretch the vectors or move their direction.
• The span, \( \operatorname{Span}\{\mathbf{v}_{1}, \mathbf{v}_{2}\} \), represents all possible vectors we can create using different \( a \) and \( b \) values.

This concept is fundamental in linear algebra because it shows how vectors can create new spaces in an n-dimensional setting. It tells us which directions we can reach using these vectors.

Linear dependence between vectors happens when you can express one vector as a scalar multiple of another. Such vectors point in the same or exactly opposite direction, meaning they don't bring anything new to the set. To check if our vectors \( \mathbf{v}_{1} \) and \( \mathbf{v}_{2} \) are linearly dependent, we try to find a scalar \( k \) such that \( \mathbf{v}_{1} = k\mathbf{v}_{2} \).In this exercise, by comparing each component of the vectors, we found that \( k = \frac{2}{3} \). This means \( 8 = 12k \), \( 2 = 3k \), and \( -6 = -9k \) all hold true. Since there is a consistent scalar across all components, \( \mathbf{v}_{1} \) can indeed be transformed into \( \mathbf{v}_{2} \) using this \( k \).When vectors are linearly dependent:

• They don't cover more dimensions. If they were in \( \mathbb{R}^3 \), their span remains in a line rather than a plane or a full 3D space.
• Adding more vectors that are dependent on each other doesn't expand the span, they stay constrained.

Geometric representation transforms abstract vector concepts into visual ones. For vectors that reside in \( \mathbb{R}^3 \), their geometric spans help illustrate what portion of the 3D space they cover.In our specific problem, the span of \( \mathbf{v}_{1} \) and \( \mathbf{v}_{2} \) turns out to be a line. Since \( \mathbf{v}_{1} \) and \( \mathbf{v}_{2} \) are linearly dependent, one vector is just a longer or shorter arrow of the other. Thus, they trace out a line rather than expanding into more dimensions like a plane.To visualize:

• Imagine the original vector \( \mathbf{v}_{1} \) is pointing in a direction in space.
• The span, which adds linear combinations of \( \mathbf{v}_{1} \) and \( \mathbf{v}_{2} \), will cover every point along the line that \( \mathbf{v}_{1} \) traces.
• Even though we live in a 3D world, the span here is restricted to a 1D line within that 3D space.

Geometric representations are crucial in helping us understand such spans as they offer a tangible way of visualizing the abstract nature of vectors and their potential combinations.