Linear Combination Calculator (find c₁, c₂, … with c₁v₁ + c₂v₂ + … = b)

Understanding the linear combination calculator — concept and principle

This calculator decides whether a target vector b can be written as a linear combination c₁v₁ + c₂v₂ + … of given vectors and, if so, finds the coefficients. It reports whether the coefficients are unique and the rank of the vector set. Use it for linear algebra homework on span, consistency and bases.

How it is calculated

Build the augmented matrix [v₁ v₂ … v_k | b] with the vectors as columns.

Row-reduce it:

• A pivot in the last column means the system is inconsistent: b is not in the span.
• Otherwise b is in the span, and the solution gives c₁, c₂, …
• The coefficients are unique when every vector column has a pivot (rank = number of vectors); otherwise there are infinitely many choices.

The principle

Writing b = c₁v₁ + … + c_kv_k component by component gives a system of linear equations with the c's as unknowns. So "is b a combination of these vectors?" is the same question as "does this system have a solution?", and also "is b in the column space?". Row reduction answers all three at once. If the vectors are independent, each reachable b has exactly one recipe.

Worked example

Default inputs: v₁ = (1, 2, 1), v₂ = (0, 1, 2), v₃ = (1, 0, −1), b = (2, 5, 4).

1. Augmented matrix [1 0 1 | 2; 2 1 0 | 5; 1 2 −1 | 4].
2. Row reduction gives a pivot in every vector column and none in the last column: rank 3.
3. Yes — b is in the span, with coefficients 2, 1, 0.
4. b written out: 2·v₁ + v₂ = (2, 4, 2) + (0, 1, 2) = (2, 5, 4).
5. Unique coefficients? yes.

How to read the result

With three independent vectors in ℝ³, the span is all of ℝ³, so every b is reachable and the answer is always unique. A coefficient of 0, as for v₃ here, simply means that vector is not needed for this particular b. If the rank were less than the number of vectors, some vectors would be redundant and the coefficients would not be unique.

Common mistakes

• Entering the vectors as rows of the coefficient matrix instead of columns.
• Concluding "not in the span" from an arithmetic slip; check by substituting.
• Assuming unique coefficients without checking the rank.
• Mixing vectors of different lengths.

Frequently asked questions

What is a linear combination?

A sum of vectors each multiplied by a scalar: c₁v₁ + c₂v₂ + …

How do I check if b is in the span?

Row-reduce [v₁ … v_k | b]; b is in the span if there is no pivot in the last column.

When are the coefficients unique?

When the vectors are linearly independent.

Sources: D. Margalit & J. Rabinoff, Interactive Linear Algebra, §2.7 Basis and Dimension · A. Davis, P. Zachlin et al., Coordinated Linear Algebra (University of Nebraska–Lincoln), §2.4