# Homework Help: Linearly independence of vector function

1. Jun 29, 2013

### HAMJOOP

Given two vectors
x(t) = (e^t te^t)^T

y(t) = (1 t)^T

a) Show that x and y are linearly dependent at each point in the interval [0, 1]

b) Show that x and y are linearly independent on [0, 1]

I compute det([x y]) = 0, so they are linearly dependent

The above problem comes from Elementary Differential Equations and Boundary Value Problems 9th ed.

Another question
given two vectors depends on t, v and w each has two components

det([v w]) = 0 at some points only
Can I say v and w are linearly dependent at those points ??

2. Jun 29, 2013

### jbunniii

In part (a), you are fixing a value of $t$, call it $t = t_0$, so the elements of the vectors are simply numbers. The linear dependence means that there exist coefficients $a$ and $b$ such that $a x(t_0) + b y(t_0) = 0$. But the coefficients will vary with $t_0$.
Part (b) is asking you to show that there are no coefficients $a$ and $b$ for which $ax(t) + by(t) = 0$ is true simultaneously for all $t \in [0,1]$.