Determine whether the given functions are linearly...

Determine whether the given functions are linearly independent or linearly dependent: (a) $2-x^2, 3 x, x^2+x-2$, (b) $3 x-1, x(2 x+1), x(x-1) ;$ (c) $e^x, e^{x+1}$; (d) $\sin x$, $\sin (x+1) ;(e) e^x, e^{x+1}, e^{x+2} ;$ (f) $\sin x, \sin (x+1), \sin (x+2) ;(g) e^x, x e^x, x^2 e^x$; (h) $e^x, e^{2 x}, e^{3 x}$; (i) $x+y, x-y+1, x+3 y+2$ - these are functions of two variables.

Determine whether the given functions are linearly independent or linearly dependent:
(a) $2-x^2, 3 x, x^2+x-2$, (b) $3 x-1, x(2 x+1), x(x-1) ;$ (c) $e^x, e^{x+1}$; (d) $\sin x$, $\sin (x+1) ;(e) e^x, e^{x+1}, e^{x+2} ;$ (f) $\sin x, \sin (x+1), \sin (x+2) ;(g) e^x, x e^x, x^2 e^x$;
(h) $e^x, e^{2 x}, e^{3 x}$; (i) $x+y, x-y+1, x+3 y+2$ - these are functions of two variables.

Key Concepts

Linear Independence

This concept refers to a set of functions (or vectors) where no nontrivial linear combination can yield the zero function (or zero vector). In other words, the only solution to c1*f1 + c2*f2 + ... + cn*fn = 0 is when all scalar coefficients c1, c2, ..., cn are zero. This idea is fundamental in understanding bases and dimensions within vector spaces, including those comprised of functions.

Linear Dependence

In contrast to linear independence, a set of functions is linearly dependent if there exists a nontrivial combination — at least one of the coefficients is nonzero — such that the sum of these weighted functions equals the zero function. This indicates redundancy in the set, meaning at least one of the functions can be expressed as a combination of the others, reducing the effective dimension of the space spanned by the set.

Vector Spaces of Functions

Functions can be considered elements of a vector space where operations like function addition and scalar multiplication are defined. This perspective allows one to apply the principles of linear algebra, such as linear independence, basis formation, and dimension, to analyze sets of functions, whether they are polynomials, exponentials, trigonometric functions, or others.

Wronskian Test

For sets of differentiable functions, the Wronskian provides a determinant-based method to assess linear independence. By constructing a matrix composed of functions and their derivatives and evaluating its determinant, one can determine if the functions are linearly independent over a particular interval. A nonzero Wronskian typically implies that the functions do not share a linear dependency.

Coefficient Comparison

When dealing with functions like polynomials or exponentials, one effective method to check for linear independence is to set up a linear combination equal to the zero function and compare the coefficients of like terms. By equating these coefficients, a system of equations can be generated; if the only solution is the trivial one (all coefficients zero), the set is linearly independent.

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