Question

14) V is a vector space. Mark each statement True or False. a. The number of pivot columns of a matrix equals the dimension of its column space. b. A plane in R' is a two-dimensional subspace of R'. c. The dimension of the vector space P, is 4. d. If dim V = n and S is a linearly independent set in V. then S is a basis for V. e. If a set fv.....v} spans a finite-dimensional vector space V and if T is a set of more than p vectors in V. then T is linearly dependent.

Answer 1

Aya) True. - If there is that means there mahrin which are dim (column space n number of ploot columns. are n columns in that linearing independent. Hence, the matrix !=n. of b) False Consider amy plane which does not pass through the origin in IR². That means 5 = (0,0,0) does not belong to the plane. Hence it connot be a subspace in IR², con any subspace must contain o element) c) False Consider the linearly independen set in IPA. s={1,x, x2, x3, x4} . C cleanly s is linearly independent, on no element can be expressed on combination of others) a. dim CIPA ) >s. so, the statement is false, false dim (V)=n. fits is lineonly independent set with less than an elements. Then it commot span whole v. so s connot be basis of v. e) Tone A set {u, 2 – r., & spans V.

therefore, every element can be written an linear combination of these l elements Now, I contain more than p elements, there can be atmost p elements in T which one linearly independent. So, this means e element in T, which can be generated from that p elements, this leads to the fact that is dependent set. ( lineonly)