ENGG2013 Unit 14

Subspace and dimensionMar, 2011.

Yesterday

• Every basis in contains two vectors

• Every basis in contains three vectors

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x

y

xy

z

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

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Cities in China

Shanghai

Beijing

Hong Kong

Tianjing

Wuhan

Guangzhou

Shenzhen

Subset of cities in Guangdongprovince

Review: Intersection and union

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F: Set of fruits

A: subset o

f fruit w

ith re

d skin

B: seedless

A union B = {cherry, apple, raspberry, watermelon}

A intersect B = {raspberry}

Subspace: definition

• A subspace W in is a subset which is– Closed under addition– Closed under scalar multiplication

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W

Conceptual illustration

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W

Example of subspace

• The z-axis

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x

y

z

Example of subspace• The x-y plane

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x

y

z

Non-example

• Parabola

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x

y

Intersection

• Intersection of two subpaces is also a subspace.

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x

y

z

For example, the intersectionof the x-y plane and the x-z planeis the same as the x-axis

Union • Union of two subspace is in general not a subspace.

– It is closed under scalar multiplicationbut not closed under addition.

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x

y

z

For example, the unionof the x-y plane and the z axisis not closed under addition

Lattice points

• The set is not a subspace

– It is closed under addition, – But not closed under scalar multiplication

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1

1

2

2

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 , if1.The k vectors are linearly independent2.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

• 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.

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x

yz

W

Example

• Let W be the y-axis

• The set

containing only one elementis a basis of W.

Dimension of W is 1.kshum ENGG2013 17

x

y

z

W

Question

• Let W be the y-axisshifted to the right by one unit.

• What is the dimensionof W?

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x

yz

W1

Question

• Let W be the straight line x=y=z.

• What is the dimension of W?

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Question

• Find a basis for the plane

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Question

• Find a basis for the intersection of

(This is the intersection of two planes:x – 2y – z = 0, and x + y + z = 0.)

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