MAT1341-L05-Subspaces1

MAT1341-L05-Subspaces1 - SUBSPACES AND SPANS JOS ´ E MALAG...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

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

Unformatted text preview: SUBSPACES AND SPANS JOS ´ E MALAG ´ ON-L ´ OPEZ In a vector space we have two basic operations: addition and scalar multiplication. Linear algebra is about the study of the objects that are completely described in terms of such operations. Specifically, we will pay attention to: 1) all the elements in a vector space V that can be written in terms of the two basic operations, which leads to the notion of linear combination; 2) all the subsets of a vector space V that inherit a structure of vector space from the structure given in V , which leads to the notion of subspace. Subspaces A subset W of a vector space V is a subspace if W is a vector space under the operations of addition and scalar multiplication given on V . Example. A vector space V is a subspace of V . Also, if is the zero vector of V , then { } is a subspace of V . We say that { } is the zero subspace . Remark. It is possible that a subset S of a vector space V could have a structure of a vector space under operations different from the op- erations of addition and scalar product of V . In such case, S is not a subspace of V . As an example, take the set S of all the 2 × 2 matrices of the form parenleftbigg a 1 1 b parenrightbigg where a and b are real numbers. We have that S is a vector space under parenleftbigg a 1 1 b parenrightbigg + parenleftbigg a ′ 1 1 b ′ parenrightbigg = parenleftbigg a + a ′ 1 1 b + b ′ parenrightbigg and α⋆ parenleftbigg a 1 1 b parenrightbigg = parenleftbigg αa 1 1 αb parenrightbigg . But we have that S is not a vector space under the standard operations for M 2 × 2 . Since the standard operations for M 2 × 2 are different from the operations defined above, we have that S is not a subspace of M 2 × 2 . 1 If we take a look at the axioms defining a vector space we notice that the only axioms that we need to check for a subset W of a vector space V to see if it is a subspace are: • W is closed under addition; • existence of additive identity; • existence of additive inverse; • W is closed under scalar multiplication. Now, let w be any vector in W ⊆ V . If W is closed under scalar multiplication then − w = ( − 1) w is in W . In other words, we do not need to check the existence of additive inverse in W . Summarizing, we have obtained: Theorem 1. Let V be a vector space. Let W be a non-empty subspace of V . Then W is a subspace of V if and only if (1) The zero vector in V is in W . (2) W is closed under addition. (3) W is closed under scalar multiplication. Example. R is a subspace of C . Example. The set { ( x y ) T | x + y = 0 } is a subspace of R 2 under the standard operations. Indeed, notice that • (0 0) T is in W : we just observe that (0 0) T satisfies the equa- tion defining W , 0 + 0 = 0. Thus (0 0) T ∈ W ....
View Full Document

{[ snackBarMessage ]}

Page1 / 9

MAT1341-L05-Subspaces1 - SUBSPACES AND SPANS JOS ´ E MALAG...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online