ENGG2013 Unit 13

BasisFeb, 2011.

Question 1

• Find the value of c1 and c2 such that

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Question 2

• Find the value of c1 and c2 such that

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Question 3

• Find c1, c2, c3 and c4 such that

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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 5

Example

• form a basis of .

• Another notation is:

is a basis of .

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1

1

Example

• form a basis of .

• Another notation is:

is a basis of .

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2

2

Non-Example

• is not a basis of .

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1

1

Alternate definition of basis

• A set of k vectors

is a basis of if the k vectors satisfy:1. They are linear independent2. The span of them is equal to (this is a short-hand of the statement that:

every vector in can be written as a linear combination of these k vectors.)

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More examples

• is a basis of

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3

3

Question

• Is a basis of

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11

x

y

z

Question

• Is a basis of ?

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11

x

y

z

1

Question

• Is a basis of ?

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3

2x

y

z

1

1

Question

• Is a basis of ?

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2

x

y

z

1

1

Question

• Is a basis of ?

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2

x

y

z

1

1

Fact

• Any two vectors in do not form a basis.– Because they cannot span the whole .

• Any four or more vectors in do not form a basis– Because they are not linearly independent.

• We need exactly three vectors to form a basis of .

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A test based on determinant

• Somebody gives you three vectors in .• Can you tell quickly whether they form a

basis?

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TheoremThree vectors in form a basis if and only if the determinant

obtained by writing the three vectors together is non-zero.

Proof: Let the three vectors be

Assume that they form a basis. In particular, they are linearly independent. By definition, this means

that if

then c1, c2, and c3 must be all zero.By the theorem in unit 12 (p.17) , the determinant is nonzero.

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This theorem generalizesto higher dimension naturally.Just replace 3x3 det by nxn det

The direction of the proof

• In the reverse direction, suppose that

• We want to show that1. The three columns are linearly independent2. Every vector in can be written as a linear

combination of these three columns.

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The direction of the proof1. Linear independence: Immediate from the theorem

in unit 12 (8 3).2. Let be any vector in .

We want to find coefficients c1, c2 and c3 such that

Using (8 1), we know that we can find a left inverse of . We can multiply by the left

inverse from the left and calculate c1, c2, c3.

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Example

• Determine whether form a basis.

• Check the determinant of

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Summary

• A basis of contains the smallest number of vectors such that every vector can be written as a linear combination of the vectors in the basis.

• Alternately, we can simply say that: A basis of is a set of vectors, with fewest number of

vectors, such that the span of them is .

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