To find the orthogonal projection of v v v W W W u 1 , u 2 , u 3 u_1, u_2, u_3 u 1 , u 2 , u 3
Formula for the Orthogonal Projection
The orthogonal projection of v v v W W W Proj W ( v ) = ∑ i = 1 3 Proj u i ( v ) , \text{Proj}_W(v) = \sum_{i=1}^3 \text{Proj}_{u_i}(v), Proj W ( v ) = ∑ i = 1 3 Proj u i ( v ) , u i u_i u i Proj u i ( v ) = ⟨ v , u i ⟩ ⟨ u i , u i ⟩ u i . \text{Proj}_{u_i}(v) = \frac{\langle v, u_i \rangle}{\langle u_i, u_i \rangle} u_i. Proj u i ( v ) = ⟨ u i , u i ⟩ ⟨ v , u i ⟩ u i .
Step 1: Compute Projections
Compute Proj u 1 ( v ) \text{Proj}_{u_1}(v) Proj u 1 ( v )
Compute ⟨ v , u 1 ⟩ \langle v, u_1 \rangle ⟨ v , u 1 ⟩ ⟨ v , u 1 ⟩ = 6 ( 1 ) + ( − 4 ) ( 1 ) + 8 ( 0 ) + ( − 6 ) ( 0 ) = 6 − 4 = 2. \langle v, u_1 \rangle = 6(1) + (-4)(1) + 8(0) + (-6)(0) = 6 - 4 = 2. ⟨ v , u 1 ⟩ = 6 ( 1 ) + ( − 4 ) ( 1 ) + 8 ( 0 ) + ( − 6 ) ( 0 ) = 6 − 4 = 2.
Compute ⟨ u 1 , u 1 ⟩ \langle u_1, u_1 \rangle ⟨ u 1 , u 1 ⟩ ⟨ u 1 , u 1 ⟩ = 1 2 + 1 2 + 0 2 + 0 2 = 1 + 1 = 2. \langle u_1, u_1 \rangle = 1^2 + 1^2 + 0^2 + 0^2 = 1 + 1 = 2. ⟨ u 1 , u 1 ⟩ = 1 2 + 1 2 + 0 2 + 0 2 = 1 + 1 = 2.
Compute Proj u 1 ( v ) \text{Proj}_{u_1}(v) Proj u 1 ( v ) Proj u 1 ( v ) = ⟨ v , u 1 ⟩ ⟨ u 1 , u 1 ⟩ u 1 = 2 2 [ 1 1 0 0 ] = [ 1 1 0 0 ] . \text{Proj}_{u_1}(v) = \frac{\langle v, u_1 \rangle}{\langle u_1, u_1 \rangle} u_1 = \frac{2}{2} \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}. Proj u 1 ( v ) = ⟨ u 1 , u 1 ⟩ ⟨ v , u 1 ⟩ u 1 = 2 2 1 1 0 0 = 1 1 0 0 .
Compute Proj u 2 ( v ) \text{Proj}_{u_2}(v) Proj u 2 ( v )
Compute ⟨ v , u 2 ⟩ \langle v, u_2 \rangle ⟨ v , u 2 ⟩ ⟨ v , u 2 ⟩ = 6 ( 1 ) + ( − 4 ) ( − 1 ) + 8 ( − 1 ) + ( − 6 ) ( 1 ) = 6 + 4 − 8 − 6 = − 4. \langle v, u_2 \rangle = 6(1) + (-4)(-1) + 8(-1) + (-6)(1) = 6 + 4 - 8 - 6 = -4. ⟨ v , u 2 ⟩ = 6 ( 1 ) + ( − 4 ) ( − 1 ) + 8 ( − 1 ) + ( − 6 ) ( 1 ) = 6 + 4 − 8 − 6 = − 4.
Compute ⟨ u 2 , u 2 ⟩ \langle u_2, u_2 \rangle ⟨ u 2 , u 2 ⟩ ⟨ u 2 , u 2 ⟩ = 1 2 + ( − 1 ) 2 + ( − 1 ) 2 + 1 2 = 1 + 1 + 1 + 1 = 4. \langle u_2, u_2 \rangle = 1^2 + (-1)^2 + (-1)^2 + 1^2 = 1 + 1 + 1 + 1 = 4. ⟨ u 2 , u 2 ⟩ = 1 2 + ( − 1 ) 2 + ( − 1 ) 2 + 1 2 = 1 + 1 + 1 + 1 = 4.
Compute Proj u 2 ( v ) \text{Proj}_{u_2}(v) Proj u 2 ( v ) Proj u 2 ( v ) = ⟨ v , u 2 ⟩ ⟨ u 2 , u 2 ⟩ u 2 = − 4 4 [ 1 − 1 − 1 1 ] = [ − 1 1 1 − 1 ] . \text{Proj}_{u_2}(v) = \frac{\langle v, u_2 \rangle}{\langle u_2, u_2 \rangle} u_2 = \frac{-4}{4} \begin{bmatrix} 1 \\ -1 \\ -1 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 1 \\ 1 \\ -1 \end{bmatrix}. Proj u 2 ( v ) = ⟨ u 2 , u 2 ⟩ ⟨ v , u 2 ⟩ u 2 = 4 − 4 1 − 1 − 1 1 = − 1 1 1 − 1 .
Compute Proj u 3 ( v ) \text{Proj}_{u_3}(v) Proj u 3 ( v )
Compute ⟨ v , u 3 ⟩ \langle v, u_3 \rangle ⟨ v , u 3 ⟩ ⟨ v , u 3 ⟩ = 6 ( 0 ) + ( − 4 ) ( 0 ) + 8 ( 1 ) + ( − 6 ) ( 1 ) = 8 − 6 = 2. \langle v, u_3 \rangle = 6(0) + (-4)(0) + 8(1) + (-6)(1) = 8 - 6 = 2. ⟨ v , u 3 ⟩ = 6 ( 0 ) + ( − 4 ) ( 0 ) + 8 ( 1 ) + ( − 6 ) ( 1 ) = 8 − 6 = 2.
Compute ⟨ u 3 , u 3 ⟩ \langle u_3, u_3 \rangle ⟨ u 3 , u 3 ⟩ ⟨ u 3 , u 3 ⟩ = 0 2 + 0 2 + 1 2 + 1 2 = 1 + 1 = 2. \langle u_3, u_3 \rangle = 0^2 + 0^2 + 1^2 + 1^2 = 1 + 1 = 2. ⟨ u 3 , u 3 ⟩ = 0 2 + 0 2 + 1 2 + 1 2 = 1 + 1 = 2.
Compute Proj u 3 ( v ) \text{Proj}_{u_3}(v) Proj u 3 ( v ) Proj u 3 ( v ) = ⟨ v , u 3 ⟩ ⟨ u 3 , u 3 ⟩ u 3 = 2 2 [ 0 0 1 1 ] = [ 0 0 1 1 ] . \text{Proj}_{u_3}(v) = \frac{\langle v, u_3 \rangle}{\langle u_3, u_3 \rangle} u_3 = \frac{2}{2} \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}. Proj u 3 ( v ) = ⟨ u 3 , u 3 ⟩ ⟨ v , u 3 ⟩ u 3 = 2 2 0 0 1 1 = 0 0 1 1 .
Step 2: Add the Projections
Proj W ( v ) = Proj u 1 ( v ) + Proj u 2 ( v ) + Proj u 3 ( v ) . \text{Proj}_W(v) = \text{Proj}_{u_1}(v) + \text{Proj}_{u_2}(v) + \text{Proj}_{u_3}(v). Proj W ( v ) = Proj u 1 ( v ) + Proj u 2 ( v ) + Proj u 3 ( v ) . Proj W ( v ) = [ 1 1 0 0 ] + [ − 1 1 1 − 1 ] + [ 0 0 1 1 ] = [ 0 2 2 0 ] . \text{Proj}_W(v) = \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} -1 \\ 1 \\ 1 \\ -1 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 2 \\ 2 \\ 0 \end{bmatrix}. Proj W ( v ) = 1 1 0 0 + − 1 1 1 − 1 + 0 0 1 1 = 0 2 2 0 .
Final Answer:
The orthogonal projection of v v v W W W Proj W ( v ) = [ 0 2 2 0 ] . \text{Proj}_W(v) = \begin{bmatrix} 0 \\ 2 \\ 2 \\ 0 \end{bmatrix}. Proj W ( v ) = 0 2 2 0 .
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Related Questions:
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Tip:
Always verify the orthogonality of the u i u_i u i