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Let W be the subspace of R^4 spanned by the vectors [-1, -1, -1, 1] and [-4, -6,...
Apr 20, 2024
Let W be the subspace of R^4 spanned by the vectors [-1, -1, -1, 1] and [-4, -6, 0, 2]. Find the matrix A of the orthogonal projection onto W
Solution by Steps
step 1
Let v1=[−1,−1,−1,1]v_1 = [-1, -1, -1, 1] and v2=[−4,−6,0,2]v_2 = [-4, -6, 0, 2] be the basis vectors for the subspace WW
step 2
Construct a matrix VV with v1v_1 and v2v_2 as its columns: V=[−1amp;−4−1amp;−6−1amp;01amp;2]V = \begin{bmatrix} -1 & -4 \\ -1 & -6 \\ -1 & 0 \\ 1 & 2 \end{bmatrix}
step 3
Compute the matrix product VVTV V^T: VVT=[−1amp;−4−1amp;−6−1amp;01amp;2][−1amp;−1amp;−1amp;1−4amp;−6amp;0amp;2]V V^T = \begin{bmatrix} -1 & -4 \\ -1 & -6 \\ -1 & 0 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} -1 & -1 & -1 & 1 \\ -4 & -6 & 0 & 2 \end{bmatrix}
step 4
Simplify the product to get VVT=[17amp;26amp;4amp;−626amp;40amp;6amp;−84amp;6amp;1amp;−1−6amp;−8amp;−1amp;5]V V^T = \begin{bmatrix} 17 & 26 & 4 & -6 \\ 26 & 40 & 6 & -8 \\ 4 & 6 & 1 & -1 \\ -6 & -8 & -1 & 5 \end{bmatrix}
step 5
Compute the inverse of VVTV V^T: (VVT)−1(V V^T)^{-1}
step 6
Multiply the inverse by VV: A=V(VVT)−1VTA = V (V V^T)^{-1} V^T
step 7
The matrix AA is the orthogonal projection matrix onto WW
Answer
The matrix AA of the orthogonal projection onto WW is obtained by computing V(VVT)−1VTV (V V^T)^{-1} V^T.
Key Concept
Orthogonal Projection Matrix
Explanation
The orthogonal projection matrix onto a subspace spanned by a set of vectors can be found using the formula A=V(VVT)−1VTA = V (V V^T)^{-1} V^T, where VV is a matrix whose columns are the basis vectors for the subspace.
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