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Chapter 6: Q8E (page 326)

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

{x2,鈥�x2-1,鈥�5}on(-鈭�,鈥�鈭�)

Short Answer

Expert verified

Thus,x2,鈥�x2-1,鈥�5 are linearly dependent on -鈭�,鈥�鈭�.

Step by step solution

01

Step 1:Using the concept of Wronskian

The given function isx2,鈥�x2-1,鈥�5.

Apply the concept of Wronskian,

Wf1,f2,鈥�,fn=f1xf2x鈥�fnxf1'xf2'x鈥�fn'x鈰�鈰�鈰�f1n-1xf2n-1x鈰�fnn-1x

Therefore,

Wx2,x2-1,5=x2x2-152x2x0220

Solve the above equation,

Wx2,鈥�x2-1,鈥�5=x20-x2-10+54x-4x=0

02

Step 2:Check the linearly independent or dependent

The above function is equal to zero鈭赌x.

Therefore,x2,鈥�x2-1,鈥�5 are linearly dependent on -鈭�,鈥�鈭�.

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