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].