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Chapter 3

Chapter 3. Section 3.2 Vector Space Properties of. Vector Space Properties Let x , y , and z be vectors in a vector space W and . W has the following properties. closure (c1) (c2) Addition (a1) (a2) (a3) and for all x (a4) If then and Multiplication (m1)

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Chapter 3

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  1. Chapter 3 Section 3.2 Vector Space Properties of

  2. Vector Space Properties Let x,y, and z be vectors in a vector space W and . W has the following properties. closure (c1) (c2) Addition (a1) (a2) (a3) and for all x (a4) If then and Multiplication (m1) (m2) (m3) (m4) for all is a vector space. What is a Vector? The answer to this questions depends on who you ask. There are several ways to vectors. (Physicist-geometric) Anything that has been assigned a direction and length. (Computer Scientist-numeric) An ordered list of numbers. (Mathematician-algebraic) An element in a vector space. A vector space is a set of “vectors” that satisfy the 2 closure, 4 addition and 4 multiplication properties given to the right. The set of column matrices we called is a vector space due to the properties of adding and scalar multiplication of matrices.

  3. (c1) (a1) (a4) (a3) (c2) (m1) (m4) Algebra and Vectors The properties of a vector space are the fundamental concepts needed in order to do basic algebraic manipulations like solving equations. The example to the right show how the properties apply to solving Subspaces A subset W of vectors may or may not form a vector space. A subset W of that is itself a vector space is called a subspace of . Any subset will satisfy the inherited properties (a1), (a2), (m1), (m2), (m3), (m4). The theorem to the right shows exactly when a subset is a subspace. • Subspace Theorem • A subset W of is a subspace of if and only if W satisfies the following 3 properties: • (a3) and for all x • (c1) If then • (c2) If and then

  4. Example Show is not a subspace. The vector since Showing W is not a subspace To show W is not a subspace you need to give a specific example of how W does not satisfy one of the properties. Showing W is a subspace. To show a subset W is a subspace you need to show that W satisfies all 3 conditions of the subspaces theorem. Example Show is a subspace. 1. Show (a3): which means 2. Show (c1): If with and then and now then which means 3. Show (c2): If and with then now then which means

  5. Example Show that is or is not a subspace. This is not subspace. If and then , but Example Show that is or is not a subspace. This is subspace. 1. Show (a3): which means 2. Show (c1): If with and then 3. Show (c2): If and with then

  6. Example Let A be a matrix show that is a subspace of . (We will call this the kernel of matrix A.) 1. Show (a3): which means 2. Show (c1): If then and then which means 3. Show (c2): If and then , then which means Example Let v be a vector in , show that is a subspace of . (We call this the orthogonal space to the vector v.) 1. Show (a3): which means 2. Show (c1): If then and , then 3. Show (c2): If and then then which means

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