How to write a vector as a linear combination?

Understanding linear combinations

Multiplying a vector by a number stretches or compresses its length and reverses its direction if the number is negative. When you understand this scalar multiplication, you can master expressing one vector in terms of others.

A linear combination means scaling two or more vectors and adding them together. In a two-dimensional plane, any vector can be written as a combination of two non-parallel vectors because their different directions allow you to travel along one line and then another to reach any destination point.

Step-by-step method with numbers

Suppose you have base vectors u = (1, 2) and v = (2, 1), and you need to express target vector w = (8, 7) as a linear combination. Start by writing the relationship: (8, 7) = c₁(1, 2) + c₂(2, 1).

Next, separate the horizontal and vertical components into two algebraic equations: 8 = c₁ + 2c₂ and 7 = 2c₁ + c₂. Multiply the first equation by 2 to get 16 = 2c₁ + 4c₂. Subtract the second equation from this new equation to eliminate c₁, giving 9 = 3c₂, which means c₂ = 3.

Finally, substitute c₂ = 3 into the first equation: 8 = c₁ + 2(3), which gives c₁ = 2. Your vector combination is w = 2u + 3v.

Checking for parallel vectors and uniqueness

This method works only if the two base vectors are not parallel. Two vectors are parallel if one is a direct scalar multiple of the other, such as (2, 4) and (6, 12). If two base vectors point along the same line, their combinations can only produce vectors along that same line.

When the base vectors are non-parallel, the representation of any target vector is completely unique. There is only one pair of scalars that works, because two intersecting lines in a plane can cross at only one specific point.