In parts (a) to (d) below, mark the statement True or False. a. Two vectors are linearly dependent if and only if they lie on a line through the origin. Choose the correct answer below. O A. True. Two vectors are linearly dependent if one of the vectors is a multiple of the other. Two such vectors will lie on the same line through the origin. O B. False. Two vectors are linearly dependent if one of the vectors is a multiple of the other. The larger vector will be further from the origin than the smaller vector. OC. False. If two vectors are linearly dependent then the graph of one will be orthogonal, or perpendicular, to the other. O D. True. Linearly dependent vectors must always intersect at the origin. b. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent. Choose the correct answer below. O A. False. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly dependent. One example is a set consisting of two vectors where one of the vectors is a scalar multiple of the other vec O B. False. A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. If there are fewer vectors than entries in the vectors, then at least one of the vectors must be written as combination of the others. O c. True. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly independent. One example is a set consisting of two vectors where one of the vectors is not a scalar multiple of the othen O D. True. If a set contains fewer vectors than there are entries in the vectors, then there are less variables than equations, so there cannot be any free variables in the equation Ax = 0. c. If x and y are linearly independent, and if z is in Span{x, y), then (x, y, z) is linearly dependent. Choose the correct answer below. O A. False. Since z is in Span{x, y), z cannot be written as a linear combination of x and y. The set (x, y, z) is linearly independent. O B. True. Vector z is in Span(x, y} and x and y are linearly independent, so z is a scalar multiple of x or of y. Since z is a multiple of x or y, the set (x, y, z) is linearly dependent. OC. True. Since z is in Span{x, y), z is a linear combination of x and y. Since z is a linear combination of x and y, the set (x, y, z) is linearly dependent. O D. False. Vector z is in Span{x, y) and x and y are linearly independent, so z must also be linearly independent of x and y. The set {x, y, z} is linearly independent.