Linear Independence Through Its Defining Equation

The Defining Equation

Linear independence is decided by asking whether a particular equation has any nontrivial solution. The equation is a linear combination equal to the zero vector, and its unknowns are the scalar coefficients. This viewpoint works in \(\mathbb{R}^n\), where the equation can be separated into coordinate equations, and in other vector spaces such as spaces of polynomials, where coefficients of like terms can be compared.

Throughout this tutorial, scalars are real numbers. A finite list \((\mathbf{v}_1,\dots,\mathbf{v}_p)\) records both its entries and their order; entries may repeat. A coefficient choice is a list of real numbers \((c_1,\dots,c_p)\). The choice is trivial if every \(c_i=0\), and nontrivial if at least one coefficient is nonzero.

The definition is about the whole list, not about whether each vector is nonzero separately. The zero coefficients always give a solution to the equation. Independence says that no other coefficient choice does. Dependence can be established by displaying one nontrivial choice and checking that its combination really is zero. To establish independence, the coordinate or coefficient equations must force every coefficient to be zero.

Coordinate Equations in \(\mathbb{R}^n\)

For vectors in \(\mathbb{R}^n\), each vector is a column with \(n\) coordinates. Adding scalar multiples coordinate by coordinate turns the defining equation into \(n\) scalar equations. Solving those equations directly tests the definition; no row-reduction criterion is needed for these small examples.

The reverse situation is just as direct: if the equations admit a nonzero coefficient choice, the list is dependent. A single such choice is enough; it is not necessary to describe every solution.

Testing Polynomial Lists

Polynomials form a real vector space under the usual addition and scalar multiplication. In the space of polynomials of degree at most \(d\), equality means that the coefficients of each power \(1,x,\dots,x^d\) agree. Thus the defining equation for a polynomial list becomes a system of scalar equations, one for each power. Powers whose coefficients are all zero contribute equations such as \(0=0\), which impose no restriction.

Consequences of the Definition

Several useful facts follow directly from the defining equation. They do not require a general test for independence; each can be checked by considering what coefficients would make a zero combination.

Proof. If \(\mathbf{v}=\mathbf{0}\), then \(1\mathbf{v}=\mathbf{0}\) is a nontrivial zero combination, so the list is dependent. If \(\mathbf{v}\ne\mathbf{0}\) and \(c\mathbf{v}=\mathbf{0}\), then \(c\ne0\) would allow division by \(c\), giving \(\mathbf{v}=\mathbf{0}\), a contradiction. Therefore \(c=0\) is the only possible coefficient, and the list is independent. \(\square\)

Proof. Let \((\mathbf{v}_1,\dots,\mathbf{v}_p)\) be independent, and select any sublist \((\mathbf{v}_{i_1},\dots,\mathbf{v}_{i_q})\). Suppose \(d_1\mathbf{v}_{i_1}+\cdots+d_q\mathbf{v}_{i_q}=\mathbf{0}\). In the full list, give each selected vector its corresponding coefficient \(d_k\) and give every omitted vector coefficient \(0\). This is a zero combination of the full list. Independence of that list forces all its coefficients to be zero, so \(d_1=\cdots=d_q=0\). This proves that the selected sublist is independent. If the selected sublist is empty, it is independent by the empty-list convention. \(\square\)

In particular, a list containing \(\mathbf{0}\) is dependent: put coefficient \(1\) on that entry and \(0\) on every other entry. A list with two equal entries is also dependent: put coefficients \(1\) and \(-1\) on those entries and \(0\) elsewhere. In each case the resulting zero combination has a nonzero coefficient. These conclusions concern lists, so repeated entries are counted as separate positions.

Proof. Suppose first that the list is independent and that two coefficient lists represent the same vector: \(a_1\mathbf{v}_1+\cdots+a_p\mathbf{v}_p=b_1\mathbf{v}_1+\cdots+b_p\mathbf{v}_p\). Subtracting the right side from the left gives \((a_1-b_1)\mathbf{v}_1+\cdots+(a_p-b_p)\mathbf{v}_p=\mathbf{0}\). Independence forces \(a_i-b_i=0\) for every \(i\), so the coefficient lists are equal. Conversely, suppose every represented vector has only one coefficient list. The zero vector has the all-zero coefficient list. Any other coefficient list that represented zero would be a second, distinct representation of the zero vector. No such list exists, so the defining equation has only the trivial solution and the list is independent. \(\square\)

Common Pitfall: Nonzero Entries Do Not Guarantee Independence

A tempting but incorrect test is to check that every vector in a list is nonzero and then declare the list independent. For example, each of \((1,0)^{\top}\), \((0,1)^{\top}\), and \((1,1)^{\top}\) is nonzero, but that fact alone does not settle the defining equation. Using the first two vectors with coefficient \(1\) and the third with coefficient \(-1\) gives

The coefficient choice \((1,1,-1)\) is nontrivial, so the list is dependent. The correction is to test the combination of the entire list, not the entries one at a time. Likewise, checking only one or two coordinate equations is not enough: independence requires that all coordinate equations together force every coefficient to be zero.

The defining-equation approach gives a consistent procedure in both coordinate and polynomial spaces: write a general linear combination, set it equal to zero, and equate the relevant coordinates or polynomial coefficients. If every solution is trivial, the list is independent; if even one nontrivial solution works, it is dependent.