Determine whether each of the following statements are true or false, where all the vectors and subspaces are in R. Justify each answer. Complete parts (a) through (c). a. If {v1. V2. V3) is an orthogonal basis for W, then multiplying vs by a scalar c gives a new orthogonal basis fV1. V2. cV3) O A. False. If the scale factor is zero it will not give a new orthogonal basis ( B. False. If the scale factor is negative it will not give a new orthogonal basis ° C. True. For example, when the vectors in an orthogonal set are normalized, they are multiplied by scalars, and the set of new vectors is orthogonal 0 D. True. Multiplication by a scalar c will change the length of v3 but it will not change the relative angles between the vectors and so they will still be orthogonal b. The Gram Schmidt process produces from a linearly independent set {x1, ·%) an orthogonal set (v1, , vp} with the property that for each k, the vectors v1,., Vk span the same subspace as that spanned by x1.Xk 0 A. False. The Gram-Schmidt process does not produce an orthogonal set from a linearly independent set, it produces an orthonormal set ? B. True. For Wk-Span(x1 , , Xk). V,-Xy and some v1 , , vk where {v1 , , vh) is an orthogonal basis for Wk, vk + 1-Xy+ 1-projWkXy+ 1 is orthogonal to wk and is in Wk + 1 . Also, vk + 1 #0. Hence, {v1 , , vk + 1)s an orthogonal basis for wk + 1 hogonal, the vectors v1 C. True. For any two basesx } and , vp} where {v1 . ??} s ort , Vk span the same subspace as that 1 , spanned by x1, , X, OD. False. When given a basis for a nonzero subspace Wof R, then . n is an orthogonal basis for W. So, the vectors don't span the same subspace QR, where Q has orthonormal columns, then R-Q' A True. Since Q has orthonormal columns then Q'Q1. So Q A Q (QR) 1R -R. C. If A O B. False. Since Q has orthonormal columns then QTQ-0 So QTA+ QTGR-0 ? ?. ? D False. f Q is square, then QT-Q 1 and Rs Q 1 A. However, if Q is not square, the matnx Q True. Since Q has orthonormal columns then Q'Q . So Q"A= Q"(QR-IR R. 1 does not exist and this statement is false