Lect12 - Chapter 3. Vector Spaces Math1111 Vector Spaces...

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Unformatted text preview: Chapter 3. Vector Spaces Math1111 Vector Spaces Example Example . Show that S = a d : a , d ∈ R with the usual matrix addition and scalar multipication is a vector space. Chapter 3. Vector Spaces Math1111 Vector Spaces Example Example . Show that S = a d : a , d ∈ R with the usual matrix addition and scalar multipication is a vector space. Example . Let W = ( a 1 ) T : a is a real number . Is W with matrix addition and scalar multiplication a vector space? Chapter 3. Vector Spaces Math1111 Vector Spaces Example Example . Show that S = a d : a , d ∈ R with the usual matrix addition and scalar multipication is a vector space. Example . Let W = ( a 1 ) T : a is a real number . Is W with matrix addition and scalar multiplication a vector space? NO Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . Let W = ( a 1 ) T : a ∈ R . Define (i) ( x 1 ) T + ( y 1 ) T = ( x + y 1 ) T for any ( x 1 ) T , ( y 1 ) T ∈ W , (ii) α ( a 1 ) T = ( α a 1 ) T for any scalar α and ( a 1 ) T ∈ W . Is W a vector space with respect to these two operations? Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . Let W = ( a 1 ) T : a ∈ R . Define (i) ( x 1 ) T + ( y 1 ) T = ( x + y 1 ) T for any ( x 1 ) T , ( y 1 ) T ∈ W , (ii) α ( a 1 ) T = ( α a 1 ) T for any scalar α and ( a 1 ) T ∈ W . Is W a vector space with respect to these two operations? Yes Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . Let W = ( a 1 ) T : a ∈ R . Define (i) ( x 1 ) T + ( y 1 ) T = ( x + y 1 ) T for any ( x 1 ) T , ( y 1 ) T ∈ W , (ii) α ( a 1 ) T = ( α a 1 ) T for any scalar α and ( a 1 ) T ∈ W . Is W a vector space with respect to these two operations? Yes Theorem 3.1.1 Let V be a vector space and x ∈ V . Then (i) x = , (ii) x + y = implies y =- x (i.e. additive inverse is unique) , (iii) (- 1 ) x =- x . (i.e. The scalar multiple (- 1 ) x is the additive inverse of x .) Chapter 3. Vector Spaces Math1111 Vector Spaces Homework 2 Reading Textbook - p.121 Homework 2 Chapter 3 - Section 1 Exercises: Qn. 7, 8, 9, 10, 11, 12, 13, 14, 16. Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } . Endow S with the addition and scalar multiplication of R 2 . Show that S is a vector space with respect to these two operations. Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } . Endow S with the addition and scalar multiplication of R 2 . Show that S is a vector space with respect to these two operations....
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Lect12 - Chapter 3. Vector Spaces Math1111 Vector Spaces...

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