Parts E, F, G, H, and I (e.) Give a geometric description of span{[1 0 1], [2 1
ID: 3167920 • Letter: P
Question
Parts E, F, G, H, and I
(e.) Give a geometric description of span{[1 0 1], [2 1 0], [-1 -2 3]}
(f.) Show that the vectors e1=[1 0 0], e2=[0 1 0] and e3= [0 0 1] are in span{[1 0 1], [2 1 0], [0 0 1]}
(g.) use your work from the previous problem to explain why every vector in R3 is also in span{[1 0 1], [2 1 0], [0 0 1]}
(h.) use part (g) to explain why the 3x3 matrix equation [1 0 1][2 1 0][0 0 1] [x1 x2 x3] = [b1 b2 b3] has no solution, no matter what b1, b2, and b3 are.
(I.) use part (h) to explain how we know that the previous matrix is invertible.
Explanation / Answer
e) the span of the given vectors is a plane in R^3 which passes through these 3 points (1,0,1),(2,1,0) as well as the origin (0,0,0). The point (-1,-2,3) is the linear combination of the other 2 as follows : 3(1,0,1) - 2(2,1,0) = (-1,-2,3)
We know that given 3 points, there passes a plane through them. Here the points are (1,0,1),(2,1,0),(000). And there span is the plane passing through them.
f) clearly e3= (0,0,1) is itself one of the given vectors.
e1 = (1,0,1) - (0,0,1) = (1,0,0)
e2 = (2,1,0) - 2(1,0,0) = (0,1,0)
Hence e1,e2,e3 can be written in the span of the given 3 vectors.
g) let a be a vector in R^3.
Then a=(a1, a2, a3) = a1(1,0,0) + a2(0,1,0) + a3(0,0,1) = a1.e1 + a2.e2 + a3.e3
But e1,e2,e3 is in the span of the given 3 vectors. Hence we can appropriately replace e1,e2,e3 by these vectors in the representation of a. Hence a can be written as the span of the given 3 vectors. Hence every vector in R^3 is in the span of the given 3 vectors.
h) given b1,b2,b3 the system has a unique solution :
x1=b1 - b3 , x2 = b2 - 2b1 + 2b3 , x3 = b3.
i) the system has unique solution IF AND ONLY IF the coefficient matrix is invertible.
Since the system in (h) has a unique solution, the matrix must be invertible