Learn how coordinates express vectors as linear combinations, how span depends on the vectors chosen, and what it means for vectors to be independent or form a basis.
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Coordinates can be understood as scalars that stretch or reverse basis vectors. In the standard plane, the horizontal and vertical unit vectors are scaled by the two coordinates, and the resulting vectors are added. Other pairs of vectors can also serve as a basis: the coordinates of a vector depend on which basis is chosen. Scaling vectors and adding them is called taking a linear combination.
The span of a set of vectors is the collection of all vectors obtainable from their linear combinations. In two dimensions, two vectors that point in different directions span the whole plane; if they line up, their span is a line, while two zero vectors reach only the origin. In three dimensions, two non-aligned vectors span a plane through the origin. A third vector outside that plane expands the span to all of 3D space, while one already in the plane adds nothing new.
A vector is redundant if it can be removed without changing the span; a set with such redundancy is linearly dependent. If every vector adds a new dimension to the span, the vectors are linearly independent. A basis is defined as a linearly independent set that spans the space.
Correct answer: C. The x-coordinate contributes three units in the positive x-direction, while the negative y-coordinate contributes two units in the negative y-direction.
Correct answer: C. Coordinates describe a vector relative to the selected basis, so changing the basis can change its coordinate description even when the vector stays fixed.
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