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Vector Spaces Week #6: Vector Spaces February 18, 2013 The linear structure of Rn appears in other sets. Consider the following examples • Let M2×2 be the set of all 2 × 2 matrices with real entries. We have a notion of addition and scalar multiplication for these matrices that works just like addition and scalar multiplication for vectors in Rn. • Let P2 be the set of all polynomials of degree two or less. We have an addition operation and a scalar multiplication operation for polynomials that works much like addition and scalar multiplication for Rn. • We have scalar multiplication and vector addition for functions, so it makes sense to define the span of a set of functions. Let W := Spanfsin(t); cos(t)g. In other words, W is the set of all functions that can be written as a linear combinations of the sine function and the cosine function. • Let C(R) denote the set of continuous functions from R to R. If g and h are elements of C(R), then for all c 2 R both g + h and cg are elements of C(R). Moreover, the basic prop- erties of addition and scalar multiplication of functions work the same as addition and scalar multiplication of vectors in Rn. In some sense all of these spaces (i.e. the set with the operations together) work just like Rn with vector addition and scalar multiplication. Most of the theory in previous material relied on certain simple algebraic properties of Rn (e.g. the distributive property for scalar multiplication with vector addition). So if we had these algebraic properties for a new set with operations (e.g. polynomials) it stands to reason that we could develop similar theory. Functions Are Like Vectors Problem 6.1. We claim that the space C(R) described above has algebraic structure similar to that of Rn with vector addition and scalar multiplication. An example of a structural property that carries over is this distributive property: if r 2 R and f and g 2 C(R) then r(f + g) = rf + rg: Try to come up with other algebraic properties that carry over - as many as you can. I can think of ten, but I don't expect you to get them all. We'll put everybody's ideas together. 1 Vector Spaces Function spaces are just one in a wide variety of sets-with-operations similar to Rn. We mentioned matrices above, but the list goes on. Whenever a set-with-operations has algebraic structure like that of Rn we call it a vector space. Definition A vector space is any nonempty set V with two operations: addition and multiplication by scalars1 subject to ten axioms listed below. The axioms must hold for all ~u, ~v and w~ in V and for all r; s 2 R. We denote the addition operation by + or ⊕ and the scalar multiplication by ·. 1. The sum of any two elements ~u and ~v in V, denoted ~u + ~v, is also an element of V. 2. ~u + ~v = ~v + ~u for any ~u; ~v 2 V: 3.( ~u + ~v) + w~ = ~u + (~v + w~ ) for any ~u; ~v; w~ 2 V: 4. There is a special element ~0 2 V, called the zero element, such that ~u + ~0 = ~u for any ~u 2 V: 5. For each ~u 2 V, there is a vector −~u in V such that ~u + −~u = ~0: 6. For every r 2 R and ~u 2 V, the scalar multiple of ~u by r, denoted r · ~u, is also in V: 7. r · (~u + ~v) = r · ~u + r · ~v for any r 2 R, ~u; ~v 2 V: 8. For any r; s 2 R and ~u 2 V, we have (r + s) · ~u = r · ~u + s · ~u. 9. For any r; s 2 R and ~u 2 V, we have r · (s · ~u) = (rs) · ~u: 10. For any ~u 2 V, 1 · ~u = ~u: The ubiquity of this structure in mathematics is what makes linear algebra such an important subject. The fact that we have explored linear algebra in such an abstract manner will make it easier to transfer the results we have to new types of vector spaces. In particular, we will use the linear structure of function spaces to understand and solve certain differential equations in Math 4B. Definition A subspace, W , of a vector space V , is a subset that satisfies the following properties: • ~0 2 W • W is closed under vector addition. i.e. If ~v; w~ 2 W , then ~v + w~ 2 W . • W is closed under scalar multiplication. i.e. If r 2 R and ~v 2 W , then r · ~v 2 W . Problem 6.2. Let V be a vector space. If W is a subspace of V , then it is also a vector space. (It just may be \smaller" than V .) Definition W is a subspace of a vector space V if it is non-empty and is itself a vector space. Problem 6.3. Explain why these two definitions are the same. 1For our purposes, scalars are real numbers; however this definition also works with complex scalars 2 Problem 6.4. Based on the definition of vector space that we just discussed, determine whether or not the following sets are vector spaces. Specifically state what scalar multiplication and vector addition are. Bonus : If a set is a vector space, try to express it as the span of a collection of vectors. (See following definition.) • S = f(x; y)jy = 0g • W = f(x; y)jy = x + 1g • W = f(x; y; z)jz = x + yg • The set of 2 × 2 matrices. • The set of vectors in the first quadrant of the plane. • The set of vectors in the first and third quadrant of the plane. • The set of polynomials of degree less than or equal to two. • The set of all polynomials of exactly degree two. • The set of all polynomials, which we denote P. • Bonus : The set of all functions f : R ! R that satisfy the differential equation f 0 = 2f. Definition Let S := ff1; f2; :::; fng be a set of functions in C(R). The span of S, denoted Span(S) is the collection of all linear combinations of f1; f2; :::; fn, i.e. S = fa1f1 + a2f2 + ::: + anfnja1; a2; :::; an 2 Rg: Problem 6.5. Which of the following sets with operations are vector spaces? Explain. • The set of all polynomials p 2 P such that p(0) = 0 with the usual addition and multiplication. • (Hard.) (R; ⊕; ·)2 where R is the set of rotations of R3, ⊕ is the usual addition of matrices and · is the usual scalar multiplication for matrices. • (Hard.) (V; ⊕; ·) where V = R+, the set of positive real numbers, but with unusual operations: for any x, y 2 R+ and λ 2 R, x ⊕ y = xy and λ · x = xλ: Problem 6.6. Generate examples that illustrate linear combination, span, subspace, linear depen- dence and basis (save basis for later) in the vector space M2×2 with the usual matrix addition and scalar multiplication. 2I just drew the circle around the plus to indicate that it is different than usual addition. 3 Linear Transformations on Function Spaces Recall that linear transformations are mappings on Rn that preserve linear combinations. Since we also have a notion of linear combination for function spaces, it is natural to think about how we can extend our definition of linear transformation to mappings from one function space to another. Consider the following mappings: (1) T : C(R) !C(R); defined by (T (f))(x) = 2f(x) + x 2 2 (2) S : P2 ! P2; defined by T (ax + bx + c) = cx + bx + a 0 (3) D : P3 ! P3; defined by T (p) = p T , S and D are all mappings from one function space to another, but they don't all preserve addition and scalar multiplication. Can you tell which do and which don't? Definition. A linear transformation is a mapping T : F!G from a function space F to a function space G that preserves scalar multiplication and addition. That is, for all functions f1 and f2 in F and scalars c in R, (1) T (c f1) = c T (f1); (2) T (f1 + f2) = T (f1) + T (f2). Problem 6.7. Determine whether or not each of the mappings T , S and D above is a linear transformation on function spaces and prove your claims. Problem 6.8. For each function that is a linear combination, give its kernel. 4.
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