Answer :
To project R2 onto the subspace spanned by (1, -1) along the subspace spanned by (1, 2), use the vector (1/2, -7/2) as the projection.
The subspace spanned by (1, -1) can be represented as V = {(a, -a) | a ∈ R}, and the subspace spanned by (1, 2) can be represented as W = {(b, 2b) | b ∈ R}. To find the projection vector e, we need to calculate the orthogonal projection of (1, -1) onto W.
First, we find a vector in W that is orthogonal to (1, -1). Let's call this vector w0. To find w0, we can take any vector in W and subtract its projection onto V. Choosing (1, 2) as a vector in W, we can calculate its projection onto V using the formula:
projV(1, 2) = ((1, 2) · (1, -1)) / ((1, -1) · (1, -1)) * (1, -1) = (1/2) * (1, -1).
Subtracting the projection from (1, 2), we get:
w0 = (1, 2) - (1/2) * (1, -1) = (1/2, 5/2).
Therefore, e = (1, -1) - w0 = (1, -1) - (1/2, 5/2) = (1/2, -7/2).
So, the projection vector e that projects R2 onto the subspace spanned by (1, -1) along the subspace spanned by (1, 2) is (1/2, -7/2).
Learn more about Orthogonal Projection click here :brainly.com/question/16701300
#SPJ11
