Subspaces of Rn, Bases and Linear Independence n ~ Definition. Consider vectors ~v1; : : : ;~vr in R . An equation of the form c1~v1 + ··· + cr~vr = 0 is called a linear relation among the vectors ~v1; : : : ;~vr. If at least one of the ci is nonzero, then we call this a nontrivial linear relation among ~v1; : : : ;~vr. Definition. Vectors ~v1; : : : ;~vk are linearly independent if there are no nontrivial linear relations among them; that is, ~v1; : : : ;~vk are linearly independent if the only way to express ~0 as a linear combination c1~v1 + ··· + ck~vk is to have c1 = c2 = ··· = ck = 0. 2 13 203 223 6 37 617 677 1. Let ~v1 = 6 7, ~v2 = 6 7, and ~v3 = 6 7. 4 15 425 445 −1 7 5 (a) Are there any nontrivial linear relations among these vectors? If so, find one. (b) Are the vectors ~v1;~v2;~v3 linearly independent? (c) Let V = span(~v1;~v2;~v3). Find a minimal set of vectors that span V . (We'll call this a basis of V .) How would you describe the shape of V ? 2 1 1 1 0 −13 21 0 3 0 −23 6 2 1 4 −1 27 60 1 −2 0 17 2. Let A = 6 7 and rref(A) = 6 7 4−1 2 −7 0 45 40 0 0 1 −55 3 0 9 −1 −1 0 0 0 0 0 (a) Find a basis of im A. (b) Find a basis of ker A. 1 Definition. A (linear) subspace of Rn is a subset V of Rn that is closed under addition and scalar multiplication; that is, • If ~v1 and ~v2 are in V , then ~v1 + ~v2 must be in V as well. • If ~v is in V and λ is any scalar, then λ~v is in V as well. (note that, ~0 must be in V because λ can be 0!) Examples: Span of vectors, kernel of A, image of A...etc. 3. Decide whether each of the following planes is a linear subspace. If it is, find a basis of the subspace. x x (a) Let V = 2 2 : x ≥ 0 . (In words, V is the set of vectors in 2 with x ≥ 0.) Is V a y R y R subspace of R2? Why or why not? x (b) Let V = 2 2 : y = ±x . Is V a subspace of 2? Why or why not? y R R x (c) Let V = 2 2 : y = 3x . Is V a linear subspace of 2? Why or why not? y R R Let V ? = f~w j~w ? ~v; for every ~v 2 V g. (All the vectors perpendicular to V ) Is V ? a linear subspace? (d) In class, we did the line y = −2x+1. which is not a linear subspace simply because ~0 is not there. 82 3 9 < x = (e) W = 4y5 : x ≤ y ≤ z . : z ; 2x3 (f) The plane Σ consisting of all vectors 4y5 satisfying 2x − 7y + 4z = 0. z T (g) fx y z : 2x − 7y + 4z = 1g. 2x3 T Note: x y z is the same as 4y5. Here T stands for transpose, meaning rows become z columns and columns become rows. Note: notation in HW8 #2: If ~a 2 V [ W , then either ~a 2 V or ~a 2 W . If ~a 2 V \ W , then ~a 2 V and ~a 2 W . For example, let V be the x-axis and W be the y-axis. V \ W is the point (0; 0), and V [ W is the x-axis and the y-axis. Is V \ W a linear subspace? How about V [ W ? 4. What are the possible linear subspaces in R2? How about in R3? 2 7 5. True or false: If ~v1; : : : ;~v5 are linearly dependent vectors in R , then ~v5 must be in span(~v1; : : : ;~v4). 6. Let A be an n × m matrix. Is im A a subspace of Rn? Why or why not? 7. Give geometric descriptions of the following: n (a) 1 vector ~v1 in R is linearly independent () . n (b) 2 vectors ~v1;~v2 in R are linearly independent () . n (c) 3 vectors ~v1;~v2;~v3 in R are linearly independent () . (d) The span of 1 linearly independent vector in Rn is . (e) The span of 2 linearly independent vectors in Rn is . (f) The span of 3 linearly independent vectors in Rn is . (g) The smallest span of vectors is . 3 Review: Using complete sentences, write definitions of the following terms. 1. kernel 2. image 3. linearly independent 4. span 5. subspace • You should be able to state the definition of a subspace of Rn, and you should have an intuitive picture of a subspace. You should also be able to determine whether a given subset of Rn is a subspace and be able to write out a complete justification of your answer. • Key examples of subspaces are the image and kernel of a matrix. • You should understand the terms linearly independent and linear relation. • If V is a subspace of Rn, you should understand what we mean by a basis of V . (Of course, you've already known for a while what a basis of Rn is, so the new thing here is that we're defining bases of subspaces of Rn.) 4.