# Let W be a subspace of R^n, and P the orthogonal projection onto W. Then Ker...

Let W be a subspace of R^n, and P the orthogonal projection onto W. Then Ker P is W^perp.

## Related Solutions

##### Problem 4. Let P be the orthogonal projection associated with a closed subspace S in a...
Problem 4. Let P be the orthogonal projection associated with a closed subspace S in a Hilbert space H, that is P is a linear operator such that P(f) = f if f ∈ S and P(f) = 0 if f ∈ S⊥. (a) Show that P2 = P and P∗ = P. (b) Conversely, if P is any bounded operator satisfying P2 = P and P∗ = P, prove that P is the orthogonal projection for some closed subspace...
##### 2a. Find the orthogonal projection of [9,40,-29,4] onto the subspace of R4 spanned by [1,6,5,6] and...
2a. Find the orthogonal projection of [9,40,-29,4] onto the subspace of R4 spanned by [1,6,5,6] and [5,1,5,5]. Answer choices: [2,14,-15,7] [-32,13,-10,7] [0,9,12,6] [-5,-2,3,2] [-12,0,-9,-9] [-16,20,0,4] [27,29,29,21] [-3,1,2,7] [-23,7,-3,-9] [-15,5,-15,30] 2b. Find the orthogonal projection of [17,18,-10,24] onto the subspace of R4 spanned by [2,7,1,6] and [3,7,3,4]. Answer choices: [-34,-22,-29,-34] [-6,4,-2,0] [-12,36,21,33] [3,21,-3,24] [7,-14,-12,1] [5,3,32,45] [14,32,12,11] [9,13,18,11] [20,2,-3,19] [-2,-6,1,-7]
##### Using least squares, find the orthogonal projection of u onto the subspace of R4 spanned by...
Using least squares, find the orthogonal projection of u onto the subspace of R4 spanned by the vectors v1, v2, and v3, where u  =  (6, 3, 9, 6), v1  =  (2, 1, 1, 1), v2  =  (1, 0, 1 ,1), v3  =  (-2, -1, 0, -1).
##### Let V -Φ -> W be linear. Show that ker (Φ) is a subspace of V...
Let V -Φ -> W be linear. Show that ker (Φ) is a subspace of V and Φ (V) is a subspace of W.
##### Let W be a subspace of Rn with an orthogonal basis {w1, w2, ..., wp} and...
Let W be a subspace of Rn with an orthogonal basis {w1, w2, ..., wp} and let {v1,v2,...,vq} be an orthogonal basis for W⊥. Let S = {w1, w2, ..., wp, v1, v2, ..., vq}. (a) Explain why S is an orthogonal set. (b) Explain why S spans Rn. (c) Showthatdim(W)+dim(W⊥)=n.
##### Let W be a subspace of R^n and suppose that v1,v2,w1,w2,w3 are vectors in W. Suppose...
Let W be a subspace of R^n and suppose that v1,v2,w1,w2,w3 are vectors in W. Suppose that v1; v2 are linearly independent and that w1;w2;w3 span W. (a) If dimW = 3 prove that there is a vector in W that is not equal to a linear combination of v1 and v2. (b) If w3 is a linear combination of w1 and w2 prove that w1 and w2 span W. (c) If w3 is a linear combination of w1 and...
##### Let W be a subspace of Rn. Prove that W⊥ is also a subspace of Rn.
Let W be a subspace of Rn. Prove that W⊥ is also a subspace of Rn.
##### Check the true statements below: A. The orthogonal projection of y onto v is the same...
Check the true statements below: A. The orthogonal projection of y onto v is the same as the orthogonal projection of y onto cv whenever c≠0. B. If the columns of an m×n matrix A are orthonormal, then the linear mapping x→Ax preserves lengths. C. If a set S={u1,...,up} has the property that ui⋅uj=0 whenever i≠j, then S is an orthonormal set. D. Not every orthogonal set in Rn is a linearly independent set. E. An orthogonal matrix is invertible.
##### (a) Find the matrix representation for the orthogonal projection Pr : R 4 → R 4...
(a) Find the matrix representation for the orthogonal projection Pr : R 4 → R 4 onto the plane P= span 1 -1 -1 1 -1 -1 1 1 (b) Find the distance of vector ~y = 2 0 0 4 from the plane P.
##### If T : R 3 → R 3 is projection onto a line through the origin,...
If T : R 3 → R 3 is projection onto a line through the origin, describe geometrically the eigenvalues and eigenvectors of T.