ENGG2013 Unit 14 Subspace and dimension Mar, 2011. Yesterday • Every basis in contains two vectors y x • Every basis in contains three vectors z y kshum x ENGG2013 2 Basis: Definition • For any given vector in if there is one and only one choice for the coefficients c1, c2, …,ck, such that we say that these k vectors form a basis of kshum ENGG2013 . 3 Review of set and subset Cities in China Tianjing Beijing Wuhan Shanghai Guangzhou Shenzhen Hong Kong Subset of cities in Guangdong province kshum ENGG2013 4 Review: Intersection and union A union B = {cherry, apple, raspberry, watermelon} F: Set of fruits A intersect B = {raspberry} kshum ENGG2013 5 Subspace: definition • A subspace W in is a subset which is – Closed under addition – Closed under scalar multiplication W kshum ENGG2013 6 Conceptual illustration W kshum ENGG2013 7 Example of subspace • The z-axis z y x kshum ENGG2013 8 Example of subspace • The x-y plane z y x kshum ENGG2013 9 Non-example • Parabola y x kshum ENGG2013 10 Intersection • Intersection of two subpaces is also a subspace. z y x For example, the intersection of the x-y plane and the x-z plane is the same as the x-axis kshum ENGG2013 11 Union • Union of two subspace is in general not a subspace. – It is closed under scalar multiplication but not closed under addition. z y x For example, the union of the x-y plane and the z axis is not closed under addition kshum ENGG2013 12 Lattice points • The set a subspace is not – It is closed under addition, – But not closed under scalar multiplication 2 1 1 kshum ENGG2013 2 13 Subspace, Basis and dimension • Let W be a subspace in • For any given vector in W, if there is one and only one choice for the coefficients c1, c2, …,ck, such that we say that these k vectors form a basis of W. and define the dimension of subspace W by dim(W)=k. kshum ENGG2013 14 Alternate definition • A set of k vectors is called a basis of a subspace W in , if 1.The k vectors are linearly independent 2.The span of them is W. The dimension of W is defined as k. We say that W is generated by these k vectors. kshum ENGG2013 15 Example z • Let W be the x-z plane • W is a subspace • u and v form a basis of W. • The dimension of W is 2. kshum ENGG2013 y W x 16 Example • Let W be the y-axis z y W • The set x containing only one element is a basis of W. Dimension of W is 1. kshum ENGG2013 17 Question • Let W be the y-axis shifted to the right by one unit. z y 1 W • What is the dimension of W? x kshum 18 ENGG2013 Question • Let W be the straight line x=y=z. • What is the dimension of W? kshum ENGG2013 19 Question • Find a basis for the plane kshum ENGG2013 20 Question • Find a basis for the intersection of (This is the intersection of two planes: x – 2y – z = 0, and x + y + z = 0.) kshum ENGG2013 21