Question 1 For each statement, decide whether it is always true (T) or sometimes

Question

Question 1 For each statement, decide whether it is always true (T) or sometimes sometimes false (F). In this question, u and v are non-zero vectors in R"; W is a vector space, and wi is a vector in W. 6 pts (a) The plane with normal vector u X v intersects any line with direction vector u (b) IIproj,ull - llproj,vll (c) Every basis for P2 has 3 elements. (d) If V is isomorphic to R3, then any set of 3 vectors in V is a basis. is linearly depende (f) Any vector is a steady-state vector for the Markov chain defined by the identity ma- trix.

Explanation / Answer

False. We know that a plane and a line either intersect or are parallel. Here, u x v is normal to u and also to the plane. Therefore, the given plane and the line are parallel and hence do not intersect. False. projv (u) = [(u.v)/(v.v)]v has the direction of v, while proju (v) = [(v.u)/(u.u)]u has the direction of u. True.as dim(P2) = 3. False. Not every set of 3 vectors in R3 or V is linearly independent. False. If a linear combination of w1,w2,w3 is 0, where not all the coefficients are 0, then w1,w2,w3 are linearly dependent. True. The steady state vector x satisfies the equation Ax = x . If A = I, then every vector satisfies this equation.

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