Linear Combinations and Span

Combining vectors and understanding what they can reach

Combining Vectors

In the previous chapter, we learned how to add vectors and multiply them by scalars. Now we combine both operations into something more powerful: the linear combination.

Think of it this way. You have two vectors, v⃗\vec{v} and w⃗\vec{w}. You can scale each one by any amount you choose, then add the results together. That final vector is a linear combination of v⃗\vec{v} and w⃗\vec{w}.

Interactive: Adjust the scalars to create different combinations

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1.0v⃗+1.0w⃗=(2.5,2.5)1.0\vec{v} + 1.0\vec{w} = (2.5, 2.5)

Mathematically, a linear combination of two vectors looks like this:

av⃗+bw⃗a\vec{v} + b\vec{w}

The numbers aa and bb are called scalars or weights. By varying these weights, you can reach many different points in space. The question that naturally arises: exactly which points can you reach?

The Span of Vectors

The span of a set of vectors is the collection of all possible linear combinations you can make from them. It answers the question: starting from the origin, where can these vectors take you?

For two vectors in 2D that point in different directions, their span is the entire plane. You can reach any point by choosing the right combination of scalars.

Interactive: Explore the span of two vectors

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The dots represent points you can reach with different scalar combinations. Together, v⃗\vec{v} and w⃗\vec{w} span the entire 2D plane.

span(v⃗,w⃗)={av⃗+bw⃗∣a,b∈R}\text{span}(\vec{v}, \vec{w}) = \{ a\vec{v} + b\vec{w} \mid a, b \in \mathbb{R} \}

This notation says: the span is the set of all vectors you can write as av⃗+bw⃗a\vec{v} + b\vec{w}, where aa and bb can be any real numbers. Positive, negative, fractional, zero—any values work.

When Vectors Are Redundant

Something interesting happens when one vector is just a scaled version of another. If w⃗=cv⃗\vec{w} = c\vec{v} for some scalar cc, then both vectors point along the same line.

In this case, adding w⃗\vec{w} does not help you reach new places. Any linear combination av⃗+bw⃗a\vec{v} + b\vec{w} equals av⃗+bcv⃗=(a+bc)v⃗a\vec{v} + bc\vec{v} = (a + bc)\vec{v}, which is still just a scaled version of v⃗\vec{v}. The span collapses from a plane to a line.

Interactive: See how parallel vectors only span a line

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When w⃗=cv⃗\vec{w} = c\vec{v}, both vectors point along the same line. They can only reach points on that line, not the full plane.

When vectors lie along the same line, we say they are linearly dependent. One vector can be expressed as a scalar multiple of the other—it is redundant.

Linear Independence

Vectors are linearly independent when none of them is a linear combination of the others. For two vectors, this simply means they do not point in the same (or opposite) direction.

The formal definition: a set of vectors is linearly independent if the only way to combine them to get the zero vector is by setting all scalars to zero.

a1v⃗1+a2v⃗2+⋯+anv⃗n=0⃗  ⟹  a1=a2=⋯=an=0a_1\vec{v}_1 + a_2\vec{v}_2 + \cdots + a_n\vec{v}_n = \vec{0} \implies a_1 = a_2 = \cdots = a_n = 0

If there is another way to get zero (with at least one non-zero scalar), then the vectors are dependent—one of them can be written in terms of the others.

Interactive: Compare independent vs dependent vectors

Independent vectors span the entire plane (2D)

Notice how independent vectors fill the plane, while dependent vectors are confined to a line. Independence means each vector contributes something new.

Span in Three Dimensions

In 3D, the situation becomes richer. A single vector spans a line. Two independent vectors span a plane—a flat 2D surface floating in 3D space. Three independent vectors span all of 3D space.

Interactive: Two vectors spanning a plane in 3D

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The translucent plane shows the span of v⃗\vec{v} (blue) and w⃗\vec{w} (red). The purple vector is their linear combination. Drag to rotate the view.

The translucent plane represents all points reachable by linear combinations of the two vectors. No matter how you adjust the scalars, the result always lies on that plane. To escape the plane and reach other points in 3D, you would need a third independent vector.

dim(span(v⃗1,v⃗2,…,v⃗n))≤n\text{dim}(\text{span}(\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n)) \leq n

The dimension of a span can never exceed the number of vectors, and it equals that number only when all vectors are independent. Dependent vectors reduce the dimension.

Why This Matters

Linear combinations are everywhere in mathematics and its applications. When you solve a system of linear equations, you are asking whether some target vector lies in the span of your coefficient vectors. When you work with bases in linear algebra, you are choosing a set of independent vectors whose span covers your entire space.

In machine learning, the features you use span a certain space of possible predictions. In physics, forces combine as linear combinations. In computer graphics, colors blend through weighted sums of RGB values.

Understanding span and independence gives you a geometric intuition for when a system has solutions, when vectors are redundant, and how many dimensions your data truly occupies.

Looking Ahead

Linear combinations are the heart of linear algebra. In the next chapter, we will see how matrices package multiple vectors together, and how matrix-vector multiplication is really just computing a linear combination of the matrix's columns. This insight connects abstract algebra to concrete geometry.

Key Takeaways

  • A linear combination scales vectors and adds them: av⃗+bw⃗a\vec{v} + b\vec{w}
  • The span of vectors is the set of all their linear combinations
  • Vectors are linearly independent when none is a combination of the others
  • Two independent vectors in 2D span the whole plane; in 3D they span a plane
  • Dependent vectors are redundant—they reduce the dimension of the span
  • The dimension of a span equals the number of independent vectors