which of the following sets vectors are independent?

How to use function rank to judge" sint,cost,cos(2t)" are independent vectors??

Answer 1

[Torsten]

In order to prove that sin(t), cos(t) and cos(2*t) are independent, you have to show that if

f(t) = a*sin(t) + b*cos(t) + c*cos(2*t)

for scalars a, b, c in IR is the identical null function (i.e. f(t) = 0 for all t), then a,b and c must all be zero.

So assume f is the null function.

Then the expression a*sin(t) + b*cos(t) + c*cos(2*t) will give zero especially when you insert t=0, t=pi/2 and t=pi.

See what follows for a,b and c by setting up the corresponding (3x3) linear system of equations for a, b and c and solving it - maybe by determining the rank of the coefficient matrix, if your assignment says you should do so.

[宇 周]

Thanks for your idea.Finally, i think the following code is right

syms t;

A=[sin(t),cos(t),cos(2*t)]

rank(A)

A =

[sin(t), cos(t), cos(2*t)]

ans =

1

[Torsten]

No, it's not right. I wonder what you think you have shown by this code.

Consider the three functions sin(t), cos(t) and sin(t)+cos(t).

Then you get the same result as above if you use

syms t
rank([sin(t), cos(t), sin(t)+cos(t)])

ans = 1

although the functions are dependent:

sin(t) + cos(t) - (sin(t)+cos(t)) == 0

[宇 周]

I agree with you.There are three vectors sin(t), cos(t) and cos(2*t), their dimension ie infinity, if they are linearly independent, the rank will be equal to three.

[Torsten]

The dimension of the three vectors is not infinity and such a thing as a "rank" for functions does not exist.

To determine whether the three functions span a three-dimensional vector space, you can either proceed as I suggested or - if you already heard about this in your course - use the Wronskian:

https://en.wikipedia.org/wiki/Wronskian

[宇 周]

Thank you.