Which of the below is/are not true? Suppose W is a subspace of R" and ŷ...
All vectors and subspaces are in R”. Mark each statement True or False. Justify each answer. Complete parts (a) through (e) below. a. If W is a subspace of R" and if y is in both W and wt, then y must be the zero vector. If v is in W, then projwv = Since the wt component of v is equal to v the w+ component of v must be A similar argument can be formed for the W...
Q6. Let W be the subspace of R' spanned by the vectors u. = 3(1, -1,1,1), uz = 5(–1,1,1,1). (a) Check that {uj,uz) is an orthonormal set using the dot product on R. (Hence it forms an orthonormal basis for W.) (b) Let w = (-1,1,5,5) EW. Using the formula in the box above, express was a linear combination of u and u. (c) Let v = (-1,1,3,5) = R'. Find the orthogonal projection of v onto W.
(1 point) Are the following statements true or false? ? 1. The best approximation to y by elements of a subspace W is given by the vector y - projw(y). ? 2. If W is a subspace of R" and if V is in both W and Wt, then v must be the zero vector. ? 3. If y = Z1 + Z2 , where z is in a subspace W and Z2 is in W+, then Z, must be...
Suppose that {ū1, ... , ūk} is a basis for a subspace W of R" and that the vector Ū E span{ū1, ... , ūk}. Then û = Proj, Ū = ū. True O False Suppose that W is a subspace of R" and that the vector ŪER" .Then if û = Projű we have Ilu - Oll < 110 - ūll for all vectors ū EW . That is, <- is the vector in W that is closest to...
Let w be a subspace of R", and let wt be the set of all vectors orthogonal to W. Show that wt is a subspace of R" using the following steps. a. Take z in wt, and let u represent any element of W. Then zu u = 0. Take any scalar c and show that cz is orthogonal to u. (Since u was an arbitrary element of W, this will show that cz is in wt.) b. Take z,...
0/1 pts Inooreat Question 9 Suppose W is a subspace of R" spanned by n nonzero orthogonal vectors. Explain why WR Two subspaces are the same when one subspace is a subset of the other subspace. Two subspaces are the same when they are spanned by the same vectors Two subspaces are the same when they are subsets of the same space Two subspaces are the same when they have the same dimension Incorrect 0/1 pts Question 10 Let U...
Suppose that {ūj, ..., ūk} is an orthonormal basis for a subspace W of R" and we form the matrix U = (ū; ū2 ... ük) Then the matrix P= UUT has the property that p2 = P . This follows for the following reason(s). A. We know that P= I and so P2 = 1? = I = P B. We can calculate p2 = (UUT) (UUT) = U (UTU) UT = UIUT C. Since P is a projection...
e, none of these 7. Let {1,..., up} be an orthogonal basis for a subspace W of R" and {...., } be an orthogonal basis for Wt. Determine which of the following is false. a. p+q=n b. {U1,..., Up, V1,...,0} is an orthogonal basis for R". c. the orthogonal projection of the u; onto W is 0. d. the orthogonal projection of the vi onto W is 0. e. none of these 8. Let {u},..., up} be an orthogonal basis...
True or False 1.aThe orthogonal complement of the subspace V = {x ∈ R^3 | x1 = x3} is a line. b. If A has QR factorization A = QR, then the columns of R span Col(A) c. The least squares solution, xˆ, to a linear system Ax = b is the vector that minimizes the quantity ||Axˆ −b||^2 . d. If A ∈ R n×n and x and y are vectors in R^n , then Ax · Ay =...
Exercise 12.6.3 Let V and W be finite dimensional vector spaces over F, let U be a subspace of V and let α : V-+ W be a surjective linear map, which of the following statements are true and which may be false? Give proofs or counterexamples O W such that β(v)-α(v) if v E U, and β(v) (i) There exists a linear map β : V- otherwise (ii) There exists a linear map γ : W-> V such that...