The vector u1 = (1,1,1,1), u2 = (0,1,1,1), u3 = (0,0,1,1), and u4 =(0,0,0,1) for
ID: 2937213 • Letter: T
Question
The vector 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 anarbitrary vector (a1,a2,a3,a4) in F4 as a linear combination ofu1,u2,u3, and u4.Please provide explanation with steps. Will rate Lifesaver. Ihave posted many questions like this so if you are good with thiskind of stuff please answer those as well and you will receive alllifesaver plus bonus points. Thank you.
Please provide explanation with steps. Will rate Lifesaver. Ihave posted many questions like this so if you are good with thiskind of stuff please answer those as well and you will receive alllifesaver plus bonus points. Thank you.
Explanation / Answer
QuestionDetails:LET
[A1,A2,A3,A4]=XU1+YU2+ZU3+TU4............LINEAR COMBINATION
WHERE X,Y,Z,T ARE SCALARS
[A1,A2,A3,A4]=X[1,1,1,1]+Y[0,1,1,1]+Z[0,0,1,1]+T[0,0,0,1]....SUBSTITUTINGFOR U1,U2,U3,U4.
[A1,A2,A3,A4]=[X,X,X,X]+[0,Y,Y,Y]+[0,0,Z,Z]+[0,0,0,T].......MULTIPLICATIONWITH SCALAR
[A1,A2,A3,A4]=[[X,(X+Y),(X+Y+Z),(X+Y+Z+T)]......ADDITION OFCORRESPONDING ELEMENTS.
EQUATING CORRESPONDING ELEMENTS ON EITHER SIDE
A1=X.................................................1
A2=X+Y.....PUTTING 1 IN THIS....A2=A1+Y......Y=A2-A1.....................2
A3=X+Y+Z..PUTTING 1&2 IN THIS....A3=A1+A2-A1+Z....Z=A3-A2.........................3
A4=X+Y+Z+T..PUTTING 1,2,3 IN THIS...A4=A1+A2-A1+A3-A2+T.....T=A4-A3................4
HENCE WE GET
[A1,A2,A3,A4]=A1U1+(A2-A1)U2+(A3-A2)U3+(A4-A3)U4
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