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.
1 0 2 3 1 7 1. Let ~v1 = , ~v2 = , and ~v3 = . 1 2 4 −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 ?
1 1 1 0 −1 1 0 3 0 −2 2 1 4 −1 2 0 1 −2 0 1 2. Let A = and rref(A) = −1 2 −7 0 4 0 0 0 1 −5 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 : 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 : y = ±x . Is V a subspace of 2? Why or why not? y R R
x (c) Let V = ∈ 2 : y = 3x . Is V a linear subspace of 2? Why or why not? y R R
Let V ⊥ = {~w |~w ⊥ ~v, for every ~v ∈ V }. (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.
x (e) W = y : x ≤ y ≤ z . z
x (f) The plane Σ consisting of all vectors y satisfying 2x − 7y + 4z = 0. z
T (g) {x y z : 2x − 7y + 4z = 1}.
x T Note: x y z is the same as y. Here T stands for transpose, meaning rows become z columns and columns become rows.
Note: notation in HW8 #2: If ~a ∈ V ∪ W , then either ~a ∈ V or ~a ∈ W . If ~a ∈ V ∩ W , then ~a ∈ V and ~a ∈ 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.)
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