math111_logo Linear Indpendence

2. Geometric intuition of linear independence

One vector u is linearly independent if

cu = 0c = 0.

This implication holds if and only if u0. Therefore we conclude

u is linearly independent ⇔ u0.

Two vectors u, v are linearly dependent if there are a and b, not all zero, such that au + bv = 0. If a ≠ 0, then we have u = (-b/a)v. If b ≠ 0, then we have v = (-a/b)u. Thus we conclude

Two vectors are linearly dependent ⇔ One is a scalar multiple of another.

In other words, the two vectors are parallel to each other. The opposite statement is

Two vectors are linearly dependent ⇔ The vectors are not parallel to each other.

For three vectors u, v, w to be linearly dependent, we have a, b, c, not all zero, such that au + bv + cw = 0. If a ≠ 0, then u = (-b/a)v + (-c/a)w is in the plane (or line, or even a point) spanned by v and w. In particular, the three vectors lie on the same plane (called coplanar). Similar arguments can be made in case other coefficients are nonzero. Thus we conclude

Three vectors are linearly independent ⇔ The vectors are on the same plane.

The linear independence of three vectors can be then understood by the opposite of the statement.

The discussion on the linear dependence can be extended to many vectors.

Vectors are linearly dependent ⇔ One is a linear combination of others.

Compared with this property of span, we conclude with the slogan

linear independence ⇔ no waste (direction in the span) ⇔ most efficient span.

In other words, deleting any vector will produce strictly smaller span.

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