Question

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lit khd where it came from 4. You do not need to simplify results, unless otherwise stated. 1. (20pts.) Indicate whether each of the following questions is True or False by writing the words "True" or "False" No explanation is needed. (a) If S is a set of linearly independent vectors in R" then the set S is an orthogonal set (b) If the vector x is orthogonal to every vector in a subspace W of R", then x is in the orthogonal complement of W. o) Far each vector y and each subapace W in t, the vector y-polw y io otiogonal to W (d) The Orthogonal Decompostion Theorem states that each y in R" can be written uniquely in the form y ý+z, where ý is in W and z is in the orthogonal complement of W. Furthermore, If { ui, . . . , up} is a basis of w, then y u y u ui u uy up and z = y (e) If W is a subspace of R" and if v is in both W and the orthogonal complem of W then v is the zero vector. MAS 31o5 ") If fv,Nz. v.. v.) an orthogonal basis ใน, wenntained in-den, an Vi fay a ncalar e basin oproduces a new banis Levs,w.ws.a) that e n (a) If x is in the subspace W, then x-projw x is not ero (h) A non-intersecting plane P parallel to the ay-plane is a subspace of (i) If f is a function in the vector space V of all real-valued smooth functions on the closed interval ja, b), and if F)-0 for some t in ja, b), then f is the zero vector in V. 2 a) (5 pts.) Find the distance between the vectors v-and a v3 2 3 s.) Find a unit vector in the direction of v-3

Answer 1

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