Week_7-8 - Week 7-8 5.2 Independence &...

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Unformatted text preview: Week 7-8 5.2 Independence & Dimension Definition: A set of vectors { ~v 1 ,~v 2 ,...,~v p } in R n are linearly independent if the vector equation x 1 ~v 1 + x 2 ~v 2 + ...x p ~v p = ~ has just the trivial solution. ( x 1 = 0 ,x 2 = 0 ,...,x p = 0) The set { ~v 1 ,~v 2 ,...,~v p } is linearly dependent if the above equation has non-trivial solutions. Example 1: If ~v 1 = 1 2 3 ~ v 2 = 4 5 6 ~v 3 = 2 1 , determine if { ~v 1 ,~v 2 ,~v 3 } is a linearly independent or dependent set. Solution Note: The vector equation x 1 ~v 1 + x 2 ~v 2 + x 3 ~v 3 = ~ 0 is equivalent to the matrix equation A~x = ~ 0. By placing the vectors ~v 1 ,~v 2 ,~v 3 as columns of a matrix A , we can see that the vectors are dependent because the matrix equation A~x = ~ 0 has non-trivial solutions. Invertible Matrix Theorem Let A be an n n matrix. TFAE 1. A is invertible 2. The RREF of A is I n . 3. A~x = ~ 0 has just the trivial solution. 1 4. A~x = ~ b has exactly one solution for each vector ~ b . 5. det( A ) = 0 6. A T is invertible 7. The columns of A are linearly . 8. The columns of A . 9. The columns of A . Linear Independence Proofs Prove the following: 1. { ~ } is linearly dependent 2. { ~v } , ( ~v 6 = ~ 0), is linearly independent. 3. A set of two vectors { ~u,~v } is linearly dependent iff one of the vectors is a scalar multiple of the other. 2 4. A set S = { ~v 1 ,~v 2 ,...,~v k } is linearly dependent iff at least one of the vectors can be written as a linear combination of the others. 5. If { ~x,~ y } is linearly independent, then { 2 ~x + 3 ~ y,~x- ~ y } is linearly independent. 6. If { ~v 1 ,~v 2 ,...,~v k } is linearly independent, and ~v / span { ~v 1 ,~v 2 ,...,~v k } , then { ~v,~v 1 ,~v 2 ,...,~v k } is linearly independent. 3 This last result will be useful in creating a largest linearly independent set of vectors by adding vectors that do not lie in the span of the original set. Basis and Dimension: A set of vectors { ~v 1 ,~v 2 ,...,~v p } in R n is called a basis for a subspace W of R n if they are linearly independent and span W . The number of vectors in a basis for W is called the dimension of W . A basis is the most efficient way to describe a subspace. Not too little, not too much. It is the smallest set of vectors that still spans the subspace and its the largest set that is still linearly independent. In other words, if we removed a vector, the remaining vectors wouldnt span and if we added another vector, the remaining vectors would no longer be independent. Examples: 1. The columns of the n n identity matrix I n form a basis for R n . This set of vectors { ~e 1 ,~e 2 ,...~e n } is called the standard basis for R n and since n vectors form a basis for R n , we say R n has dimension n ....
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Week_7-8 - Week 7-8 5.2 Independence &...

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