34.6k views
5 votes
Find The Orthogonal Projection Of 0 0 V = -5 Onto The Subspace V Of R4 Spanned By -1 1 -1 1 1 And -1 -1 1 -1 Projv(V) =

1 Answer

1 vote

Answer:

The answer is:

The orthogonal projection of v = [0 0 -5] onto the subspace V spanned by u₁ = [-1 1 -1 1] and u₂ = [-1 -1 1 -1] is projᵥ(V) = [0 0 0 0].

Explanation:

To find the orthogonal projection of the vector v = [0 0 -5] onto the subspace V of R⁴ spanned by [-1 1 -1 1] and [-1 -1 1 -1], we can use the formula for orthogonal projection:

projᵥ(V) = ((v⋅u₁)/(u₁⋅u₁)) * u₁ + ((v⋅u₂)/(u₂⋅u₂)) * u₂

where v is the vector to be projected and u₁, u₂ are the basis vectors of the subspace V.

Given:

v = [0 0 -5]

u₁ = [-1 1 -1 1]

u₂ = [-1 -1 1 -1]

Now, we can plug in the values into the formula:

projᵥ(V) = (([0 0 -5]⋅[-1 1 -1 1])/([-1 1 -1 1]⋅[-1 1 -1 1])) * [-1 1 -1 1] + (([0 0 -5]⋅[-1 -1 1 -1])/([-1 -1 1 -1]⋅[-1 -1 1 -1])) * [-1 -1 1 -1]

First, let's calculate the dot products:

([0 0 -5]⋅[-1 1 -1 1]) = (0*-1 + 0*1 + -5*-1 + 0*1) = 5

([-1 1 -1 1]⋅[-1 1 -1 1]) = (-1*-1 + 1*1 + -1*-1 + 1*1) = 4

([0 0 -5]⋅[-1 -1 1 -1]) = (0*-1 + 0*-1 + -5*1 + 0*-1) = -5

([-1 -1 1 -1]⋅[-1 -1 1 -1]) = (-1*-1 + -1*-1 + 1*1 + -1*-1) = 4

Now, let's calculate the projection:

projᵥ(V) = (5/4) * [-1 1 -1 1] + (-5/4) * [-1 -1 1 -1]

Simplifying the equation gives:

projᵥ(V) = [-5/4 5/4 -5/4 5/4] - [-(5/4) -(5/4) -(5/4) -(5/4)]

Combining like terms:

projᵥ(V) = [-5/4 + 5/4 - 5/4 + 5/4] = [0 0 0 0]

Therefore, the orthogonal projection of v = [0 0 -5] onto the subspace V is projᵥ(V) = [0 0 0 0].

User AboQutiesh
by
8.4k points