Linear Dependence and Independence¶

Definition¶

A collection $(\boldsymbol{v}_1,\boldsymbol{v}_2,\ldots,\boldsymbol{v}_k)$ of vectors is linearly dependent if there exist scalars $\alpha_1,\ldots,\alpha_k$ , not all zero, such that

\alpha_1\boldsymbol{v}_1+\alpha_2\boldsymbol{v}_2+\cdots+\alpha_k\boldsymbol{v}_k=\boldsymbol{0}.

The collection is linearly independent if this can only happen when

\alpha_1=\alpha_2=\cdots=\alpha_k=0.

Interpretation¶

Linear dependence means that at least one vector is redundant because it can be written using the others. Linear independence means that no vector in the collection is redundant: the only combination that gives $\boldsymbol{0}$ is the one with all coefficients zero.

Two panels: two vectors v1 and v2 pointing in different directions, which are independent, and three vectors where v3 = v1 + v2, which are dependent.

In the left panel, $\boldsymbol{v}_1$ and $\boldsymbol{v}_2$ point in different directions, so $\alpha_1\boldsymbol{v}_1+\alpha_2\boldsymbol{v}_2$ can only return to $\boldsymbol{0}$ when $\alpha_1=\alpha_2=0$ $0$ . In the right panel, $\boldsymbol{v}_3=\boldsymbol{v}_1+\boldsymbol{v}_2$ is the diagonal of the parallelogram, so $\boldsymbol{v}_1+\boldsymbol{v}_2-\boldsymbol{v}_3=\boldsymbol{0}$ is a combination with nonzero coefficients $1$ , $1$ and $-1$ , and the three vectors are dependent.

Drag $\boldsymbol{v}_2$ onto the line through $\boldsymbol{v}_1$ (or press the button): then a combination with coefficients not all zero gives $\boldsymbol{0}$ , and the collection is dependent.

The two readings agree. If $\alpha_1\boldsymbol{v}_1+\cdots+\alpha_k\boldsymbol{v}_k=\boldsymbol{0}$ with some $\alpha_j\ne0$ $0$ , then dividing by $\alpha_j$ , which is possible because the scalars come from a field, writes $\boldsymbol{v}_j$ as a combination of the others. Conversely, writing $\boldsymbol{v}_j$ as a combination of the others gives such a relation with coefficient $-1$ at $\boldsymbol{v}_j$ . Dependence does not mean that every vector is redundant: in $((1,0),(2,0),(0,1))$ , the vector $(0,1)$ is not a combination of the others, yet the list is dependent because $(2,0)=2(1,0)$ $2 (1, 0)$ .

Independence is what makes representations unique: if $\displaystyle \sum_i\alpha_i\boldsymbol{v}_i=\sum_i\beta_i\boldsymbol{v}_i$ $i \sum β_{i} v_{i}$ , then $\displaystyle \sum_i(\alpha_i-\beta_i)\boldsymbol{v}_i=\boldsymbol{0}$ $0$ , so $\alpha_i=\beta_i$ for every $i$ . This is why a basis must be independent. Two vectors are dependent if and only if one is a scalar multiple of the other, and three vectors in $\mathbb{R}^3$ are dependent if and only if they lie in one plane through the origin.

Key Results¶

A collection is linearly independent if and only if every vector in its span has a unique expression as a linear combination of the collection.
Any collection containing the zero vector, a repeated vector, or a scalar multiple of another vector is linearly dependent.

Examples¶

In $\mathbb{R}^2$ , $(1,0)$ and $(0,1)$ are linearly independent.
In $\mathbb{R}^2$ , $(1,0)$ and $(2,0)$ are linearly dependent, since $(2,0)=2(1,0)$ $2 (1, 0)$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 2
Math V: Advanced Linear Algebra · Week 5 (revisited)

Keywords: linearly independent, linearly dependent, linear dependence, linear independence, independent vectors, dependent vectors Symbols: alpha, beta, real numbers, sum

Last updated: 20 October 2026 03:02