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Vector Spaces

P Vanchinathan

February 8, 2012

P Vanchinathan () Vector Spaces February 8, 2012 1 / 4

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Linear DependenceA set of vectors is said to be linearly dependent if any one of them is alinear combination of the other vectors.Otherwise we call them linearly independent.

Suppose that v1 is a linear combination ofv2, bnen

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Basis of a Vector Space

A maximal set of linearly independent vectors of a vector a space is calleda basis for that vector space.

The number of elements in such a set is called the dimension of the vectorspace. Clearly R2 2-dimensional, R3 is 3-dimensional, and Rn isn-dimensional.

For any set of initial vectors form all possible linear combinations, using allscalars. The set obtained will be a vector space. (Because linearcombinations of new vectors can again be simplified to a linearcombination of origianl initial vectors).This vector space is called the span of the vectors we started with.For linearly independent set of vectors two different linear combinations

will result in two different vectors (uniqueness).Find two ways of forming linear combination giving the same result in eachof the following cases:

1 (1, 7, 1), (2,2, 3), (4, 20,72 (1, 1, 1), (3,8, 2), (0,11,1)3 (

4,

3,

2), (2, 0, 5), (10, 6, 9)P Vanchinathan () Vector Spaces February 8, 2012 4 / 4

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Rank of a MatrixIf we are given three vectors all size 3, then we can make a square matrixof order 3.Theorem: Given n vectors in Rn, they are linearly dependent if and onlyif the square matrix obtained from them is singular (i.e. has determinantzero).Example: (1, 8, 2)T, (0,4, 5)T, (3, 2, 3)T Then form matrix

1 0 3

8

4 22 5 3

. Its determinant is 1(

12

10)

0 + 2(0 + 12) = +2 so

they are linearly independent.When a matrix has determinant zero, the columns are linearly dependentvectors. What is the maximal number of columns that can be linearly

independent? That number is called the rank of the matrix.Rank is equal to the order of the largest square submatrix that hasnon-zero determinant. This is equal to the number of basic variables inthe linear system, which is the number of leading 1s in the echelon form.And the number of free variables is called the nullity of the marix.

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