(1) Suppose you are given three vectors V1,v2,V3 ? R5 and are asked to determine
ID: 3186172 • Letter: #
Question
(1) Suppose you are given three vectors V1,v2,V3 ? R5 and are asked to determine whether b ER is a linear combination of V1, V2, V3 a) Explain how you would use the definition of linear combination to write down a system of linear equations which would allow you to answer this question. Under what conditions would the system be homogeneous/inhomogeneous? b) Explain how you could obtain a matrix from this system of equations. (c) Explain how you would extract the answer to the original question from the redurxxl ?1k'Ion forul ofthe mai rix (alter Gangaaln elimination)Explanation / Answer
(1) Let v1 = (x1,x2,x3,x4,x5)T,v2 = (y1,y2,y3,y4,y5)T and v3 = (z1,z2,z3,z4,z5)T. Also, let b = (b1,b2,b3,b4,b5) .
(a) If b is a linear combination of v1,v2,v3, then there exist scalars a1,a2,a3 , not all zero, such that b = a1v1 + a2v2+a3v3 . Then, a1(x1,x2,x3,x4,x5)T+a2(y1,y2,y3,y4,y5)T+a3 (z1,z2,z3,z4,z5)T=(b1,b2,b3,b4,b5)T so that a1x1+a2y1 + a3z1 = b1…(1), a1x2+a2y2 + a3z2 = b2…(2), a1x3+a2y3 + a3z3 = b3…(3), a1x4+a2y4 + a3z4 = b4…(4) and a1x5+a2y5 + a3z5 = b5…(5).
This is the required system of linear equations.
When b1,b2,b3,b4,b5 are all zero, then the above system of linear equations is homogeneous. If any or some of b1,b2,b3,b4,b5 is/are non-zero, then the above system of linear equations is inhomogeneous.
(b). The above system of linear equations may be represented in matrix form as AX = b, where X =
x1
x1
x3
x4
x5
y1
y2
y3
y4
y5
z1
z2
z3
z4
z5
A = (a1,a2,a3) and b = (b1,b2,b3,b4,b5) .
( c). Here A is a 1x3 matrix, X is a 3x5 matrix so that AX is a 1X5 vector/matrix . Also b is a 1x5 vector/matrix. To solve this system of equations, we will reduce the augmented matrix
x1
x1
x3
x4
x5
y1
y2
y3
y4
y5
z1
z2
z3
z4
z5
b1
b2
b3
b4
b5
This will give us a solution to the system Ax = b.
x1
x1
x3
x4
x5
y1
y2
y3
y4
y5
z1
z2
z3
z4
z5
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