How to Determine Whether a Set Spans ℝ²

To determine whether a set of vectors spans $\mathbb{R}^2$, check whether the vectors can produce every point in the plane by taking linear combinations. The specific vectors are not included in the question here, so the method below shows how to test any proposed set.

The central test is simple: the set spans $\mathbb{R}^2$ if and only if its vectors have rank 2. For exactly two vectors, you can use a determinant instead of row reduction.

What it means to span $\mathbb{R}^2$

A set $S=\{v_1,\ldots,v_k\}$ spans $\mathbb{R}^2$ if every vector $(x,y)$ in the plane can be written as a linear combination of the vectors in $S$:

$$(x,y)=c_1v_1+\cdots+c_kv_k$$

for some real numbers $c_1,\ldots,c_k$. Geometrically, the vectors must provide two independent directions. If all the vectors lie on a single line through the origin, their linear combinations stay on that line and cannot fill the plane.

The rank test for any number of vectors

Write each vector as a column of a matrix. If the set contains $k$ vectors in $\mathbb{R}^2$, the matrix has two rows and $k$ columns. Row-reduce it. The set spans $\mathbb{R}^2$ precisely when the reduced matrix has a pivot in each of its two rows—that is, when its rank is 2.

This works whether the set has two vectors or more. A set with more than two vectors can still span the plane, but some of its vectors may be redundant. A set with fewer than two vectors cannot span $\mathbb{R}^2$.

The determinant test for two vectors

If the set has exactly two vectors, write them as

$$v_1=\begin{pmatrix}a\\b\end{pmatrix},\qquad v_2=\begin{pmatrix}c\\d\end{pmatrix}.$$

Place them in the columns of a matrix and calculate its determinant:

$$A=\begin{pmatrix}a&c\\b&d\end{pmatrix},\qquad \det(A)=ad-bc.$$

• If $ad-bc\ne 0$, the columns are linearly independent and span $\mathbb{R}^2$.
• If $ad-bc=0$, the columns are dependent and do not span $\mathbb{R}^2$.

The determinant is nonzero exactly when the matrix is invertible. In that case, for any target vector $w\in\mathbb{R}^2$, the equation $A\begin{pmatrix}r\\s\end{pmatrix}=w$ has a solution, so $w=rv_1+sv_2$.

Worked examples

Consider

$$v_1=\begin{pmatrix}1\\2\end{pmatrix},\qquad v_2=\begin{pmatrix}3\\1\end{pmatrix}.$$

The determinant of the matrix with these columns is

$$\det\begin{pmatrix}1&3\\2&1\end{pmatrix}=1\cdot1-3\cdot2=-5.$$

Since $-5\ne0$, the vectors span $\mathbb{R}^2$. For example, the target vector $\begin{pmatrix}7\\4\end{pmatrix}$ is their linear combination with coefficients $1$ and $2$:

$$\begin{pmatrix}7\\4\end{pmatrix}=1\begin{pmatrix}1\\2\end{pmatrix}+2\begin{pmatrix}3\\1\end{pmatrix}.$$

Now consider $u_1=\begin{pmatrix}1\\2\end{pmatrix}$ and $u_2=\begin{pmatrix}2\\4\end{pmatrix}$. Here $u_2=2u_1$, and the determinant is $1\cdot4-2\cdot2=0$. Every linear combination lies on the line $y=2x$, so these vectors do not span the plane. In particular, no combination of them equals $\begin{pmatrix}1\\0\end{pmatrix}$.

For a set of three vectors, use rank. For example, take $v_1=\begin{pmatrix}1\\0\end{pmatrix}$, $v_2=\begin{pmatrix}0\\1\end{pmatrix}$, and $v_3=\begin{pmatrix}1\\1\end{pmatrix}$. The matrix is

$$\begin{pmatrix}1&0&1\\0&1&1\end{pmatrix}.$$

It has rank 2, so the set spans $\mathbb{R}^2$. The third vector is redundant because it is $v_1+v_2$; the first two already span the plane.

Common mistakes

• Checking only whether the vectors are nonzero. Two nonzero vectors can point in the same direction, as in the second example. Nonzero vectors do not necessarily span the plane.
• Using the determinant for a non-square matrix. The determinant test applies directly only when there are exactly two vectors in $\mathbb{R}^2$. With another number of vectors, use row reduction to find the rank.
• Confusing independence with spanning. Two vectors in $\mathbb{R}^2$ span the plane exactly when they are linearly independent. With more than two vectors, the set may span even though it is dependent.
• Ignoring the zero vector. Adding the zero vector does not change a span, because it contributes nothing to any linear combination.

If you have a particular set, place its vectors in matrix columns and apply the determinant test or rank test as appropriate.