MIT18_024s11_ChAnotes

# MIT18_024s11_ChAnotes - Linear Spaces we have...

This preview shows pages 1–4. Sign up to view the full content.

Linear Spaces we have seen (12.1-12.3 of Apostol) that n-tuple space has the following properties : V, Addition: 1. (Commutativity) A + B = B + A. 2. (Associativity) A + (B+c) = (A+B) + C. 3. (Existence of zero) There is an element - 0 such 'that A + - 0 = A for all A. 4. (Existence of negatives) Given A, there is a B such that A + B = - 0. Scalar multiplication: 5. (Associativity) c (dA) = (cd)A. 6. (Distributivity) (c+d)A = cA + dA, c(A+B) = + cB. 7. (Multiplication by unity) 1A = Definition. More generally, let V be any set of objects (which we call vectors). And suppose there are two operations on as follows: The first is an operation (denoted +) that assigns to each pair A, B of vectors, a vector denoted A + B. The second is an operation that assigns to each real number c and each vector A, a vector denoted cA. Suppose also that the seven preceding properties hold. Then with these two opera- tions, is called a linear space (or a vector space). The seven properties are called the axioms -- for a linear space.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
There are many examples of linear spaces besides n--tuple space i 'n The study of linear spaces and their properties is dealt with in a subject called Linear Algebra. WE!shall treat only those aspects of linear algebra needed for calculus. Therefore we will be concerned only with n-tuple space and with certain of its subsets called "linear subspaces" : Vn - Definition. Let W be a non-empty subset of Vn ; suppose W is closed under vector addition and scalar multiplication. Then W is called a linear subspace of Vn (or sometimes simply a subspace of Vn .) To say W is closed under vector addition and scalar multiplication means that for every pair A, B of vectors of W, and every scalar c, the vectors A + B a~d cA belong to W. Note that it is automatic that the zero vector Q belongs to W, since for any A I W, we have Q = OA. Furthermore, for each A in W, the vector -A is also in W. This means (as you can readily check) that W is a linear space in its own right (i.e., f . it satisfies all the axioms for a linear,space). S~bspaces of my be specified in many different ways, as we shall see. Example 1. The subset of consisting of the 9-tuple alone is a subspace it ths "smallest possible" sub- Vn; space. Pad course V, by definition a Vn; "largest possible" subspace. W;ample 2. Let A be a fixed non-zero vector, The subset of Vn consisting of all vectors X of the form X = is a subspace of . It is called the subspace spanned by A. In the case n = 2 or 3, it can be pictured as consisting of all vectors lying on a line through the origin.
Example 3. Let A and B be given non-zero vectors that are not 1 parallel. The subset of Vn consisting of all vectors of the form is a subspace of V It is called the subspace spanned by A and no B. In the case n = 3, it can be pictured as consisting of all vectors lying in the plane through the origin that contains A and ---A/ We generalize the construction given in the preceding examples as follows: Definition. Let S = \$A?, .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/19/2012 for the course MATH 18.024 taught by Professor Christinebreiner during the Spring '11 term at MIT.

### Page1 / 35

MIT18_024s11_ChAnotes - Linear Spaces we have...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online