ENGG2013 Unit 13 Basis Feb, 2011.

Question 1 Find the value of c 1 and c 2 such that kshumENGG20132

Question 2 Find the value of c 1 and c 2 such that kshumENGG20133

Question 3 Find c 1, c 2, c 3 and c 4 such that kshumENGG20134

Basis: Definition For any given vector in if there is one and only one choice for the coefficients c 1, c 2, …,c k, such that we say that these k vectors form a basis of. kshumENGG20135

Example form a basis of. Another notation is: is a basis of. kshumENGG

Example form a basis of. Another notation is: is a basis of. kshumENGG

Non-Example is not a basis of. kshumENGG

Alternate definition of basis A set of k vectors is a basis of if the k vectors satisfy: 1.They are linear independent 2.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.) kshumENGG20139

More examples is a basis of kshumENGG

Question Is a basis of kshumENGG x y z

Question Is a basis of ? kshumENGG x y z 1

Question Is a basis of ? kshumENGG x y z 1 1

Question Is a basis of ? kshumENGG x y z 1 1

Question Is a basis of ? kshumENGG 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. kshumENGG201316

A test based on determinant Somebody gives you three vectors in. Can you tell quickly whether they form a basis? kshumENGG201317

Theorem Three 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 c 1, c 2, and c 3 must be all zero. By the theorem in unit 12 (p.17), the determinant is nonzero. kshumENGG This theorem generalizes to 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 that 1.The three columns are linearly independent 2.Every vector in can be written as a linear combination of these three columns. kshumENGG201319

The direction  of the proof 1.Linear independence: Immediate from the theorem in unit 12 (8  3). 2.Let be any vector in. We want to find coefficients c 1, c 2 and c 3 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 c 1, c 2, c 3. kshumENGG201320

Example Determine whether form a basis. Check the determinant of kshumENGG201321

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. kshumENGG201322