Question
Asked Dec 27, 2019
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The vectors u1 = (1, 1, 1, 1), u2 = (0, 1, 1, 1), u3 = (0, 0, 1, 1), and u4 = (0, 0, 0, 1) form a basis for F4.

Find the unique representation of an arbitrary vector (a1, a2, a3, a4) in F4 as a linear combination of u1, u2, u3, and u4.

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Expert Answer

Step 1

Given,

u1 = (1, 1, 1, 1), u2 = (0, 1, 1, 1), u3 = (0, 0, 1, 1), and u4 = (0, 0, 0, 1)

Consider the linear combination,

(a1, a2, a3, a4) = xu1+ yu2+zu3+tu4   …(1)

Then,

(a1, a2, a3, a4) = x (1, 1, 1, 1) + y (0, 1, 1, 1) + z (0, 0, 1, 1) + t (0, 0, 0, 1)

Now add the corresponding elements as follows,

(a1, a2, a3, a4) = [x , (x+y) , (x+y+z) , (x+y+z+t)]

Step 2

Now equate the coordinates of both sides

a1 = x

b2 = x+y …(2)

c3 = x+y+z …(3)

d4 = x+y+z+t …(4)

Substitute the value of x in (2)

b2 = a1+y implies y = b2-a1

Substitute the value of x and y in (3)

c3 = a1+(b2-a1)+z

...

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