The problem is: \"(a) What equations must a and b satisfy so that the line y = a
ID: 2983621 • Letter: T
Question
The problem is:
"(a) What equations must a and b satisfy so that the line y = ax + b passes through the points (1, 1) , (2, - 1), and ( - 1, 4)?
Write the matrix form of the system of equations (AX = b) and show that the system is inconsistent.
(b) Find the description of the column space A and show that b is not in the column space.
(c) For what values of y is the system inconsistent? For what values of y is it consistent?"
I'm just not sure how to go about doing this problem, especially part (c). So I'd greatly appreciate anyone's help. Thank you!
Explanation / Answer
The equations are
a+b = 1
2a+b = -1
-a+b = 4
In matrix form Ax=b
x is the column vector (a ; b)
b is the column vector (1 ; -1 ; 4)
A is 3 by 2 matrix (1,1 ; 2,1; -1,1)
1 1 1
2 1 -1
-1 1 4 (Augmenting b)
1 1 1
0 -1 -3
0 2 5 ( R2 -> R2 - 2R1.R3->R3+R1)
1 1 1
0 -1 -3
0 0 -1 (R3 -> R3+R2)
Look at the last row there are zeros in A for this row and nonzero in b in this row
Hence system is inconsistent
(b) column space t(1;2;-1) + u (1;1;1) = (t+u;2t+u;-t+u)
(1,-1,4) is not in the column space
because (t+u;2t+u;-t+u) = (1,-1,4) has no solution
Another interesting description of column space is via volume or determinant
column space is {(y1;y2;y3) ; det (1,1,y1; 2,1,y2 ; -1,1,y3) = 0}
You can easily check when you plug 1,-1,4 for y1,y2,y3 determinant is not zero
hence 1;-1;4 is not in column space.
This criterion comes from the fact a vector is in the plane spanned by two linearly indep vectors iff volume of parallelopiped formed by these vectors is zero
(c)Finally you can use the same determinant criterion to answer these.
System is consistent for those vectors y which when appended to vectors in A forms a matrix whose determinant is zero
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