Linear Independence and Wronskian

Related Calculator: Wronskian Calculator

A set of functions {`y_1(x)` , `y_2(x)` , ..., `y_n(x)` } is linearly dependent on `a<=x<=b` if there exist constants `c_1` , `c_2` , ... , `c_n` , not all zero, such that `c_1 y_1(x)+c_2 y_2(x)+...+c_n y_n(x)-=0` on `a<=x<=b` .

Note that `c_1`=`c_2`= ...=`c_n`= 0 is a set of constants that always satisfies given equation. A set of functions is linearly dependent if there exists another set of constants, not all zero, that also satisfies given equation. If the only solution to equation is `c_1`=`c_2`= ...=`c_n`= 0, then the set of functions {`y_1(x)` , `y_2(x)` , ..., `y_n(x)` } is linearly independent on `a<=x<=b` .

Example 1. The set {2x,x,sin(x)} is linearly dependent on [-1, 1] since there exist constants `c_1=1` , `c_2=-2` , `c_3=0` such that

`c_1*2x+c_2x+c_3sin(x)=2x-2x+0*sin(x)-=0` .

In general it is not always easy to test whether given set of functions is linearly independent through definition. Linear independence can be tested with Wronskian.

Definition. The Wronskian of a set of functions {`z_1(x)` , `z_2` (x), ..., `z_n(x)` } on the interval `a<=x<=b` , having the property

that each function possesses n-1 derivatives on this interval, is the determinant `W(z_1,z_2,...,z_n)=|[z_1,z_2,...,z_n],[z_1',z_2',...,z_n'],[z_1'',z_2'',...,z_n''],[...,...,...,...],[z_1^((n-1)),z_2^((n-1)),...,z_n^((n-1))]|`

Example 2. Wronskian of set {1,sin(x)} is `W=|[1,sin(x)],[(d(1))/(dx),(d(sin(x)))/(dx)]|=|[1,sin(x)],[0,cos(x)]|=1*cos(x)-0*sin(x)=cos(x)` .

Theorem. If the Wronskian of a set of n functions defined on the interval `a<=x<=b` is nonzero for at least one point in this interval, then the set of functions is linearly independent there. If the Wronskian is identically zero on this interval and if each of the functions is a solution to the same linear differential equation, then the set of functions is linearly dependent.

CAUTION: Theorem is silent when the Wronskian is identically zero and the functions are not known to be solutions of the same linear differential equation. In this case, one must test independence directly through definition.

Example 3. Determine whether the set {`e^x` , `cos(x)` } is linearly independent on `(-oo, oo)` .

Wronskian of this set is `W=|[e^(x),cos(x)],[(d(e^x))/(dx),(d(cos(x)))/(dx)]|=|[e^x,cos(x)],[e^x,-sin(x)]|=-e^x sin(x)-e^x cos(x)` .

Since Wronskian is nonzero for at least one point in the interval of interest, it follows from theorem that the set is linearly independent.

Theorem. The nth-order linear homogeneous differential equation L(y) = 0 always has n linearly independent solutions. If `y_1(x)` , `y_2(x)` , ..., `y_n(x)` represent these solutions, then the general solution of L(y)= 0 is `y(x)=c_1 y_1(x)+c_2 y_2(x)+...+c_n y_n(x)` where `c_1` , `c_2` , ..., `c_n` are arbitrary constants.

Example 4. Since `y_1(x)=sin(x)` and `y_2(x)=cos(x)` are solutions of differential equation `y''+y=0` and these solutions are linearly independent (because their Wronskian `W=|[sin(x),cos(x)],[cos(x),-sin(x)]|=-sin^2(x)-cos^2(x)=-1!=0` ) then the general solution of differential equation is `y(x)=c_1sin(x)+c_2cos(x)` where `c_1` and `c_2` are arbitrary constants.