Linearly independence of vector function

[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
how about part b. Isn't a) and b) are contradictory


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 ??

 

[jbunniii]

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
how about part b. Isn't a) and b) are contradictory

No, there is no contradiction.

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 $t \in [0,1]$.