3 5 Independence Basis and Dimension The true

3 -5 Independence , Basis and Dimension

• *The true dimension of the column space is the rank. • *The goal is to understand a basis : independent vectors that “ span the space ”. • *Four essential ideas: 1. Independent vectors (no extra vector) 2. Spanning a space (enough vectors to produce the rest) 3. Basis for a space (not too many or too few) 4. Dimension of a space (the number of vectors in a space)

Linear Independence • *Definition : *The columns of A are linearly independent when the only solution to Ax = 0 is x = 0. No other combination Ax of the columns gives the zero vectors. *The sequence of vectors v 1, …, vn is linearly independent if the only combination that gives the zero vector is 0 v 1+0 v 2+…+0 vn. • When the vectors go into the columns of A , Ax = 0 if and only if x = 0. • N(A) = Z

Independent or dependent • (a) The vectors (1 , 0) and (0 , 1) • (b) The vectors (1 , 0) and (0 , 0. 00001) • (c) The vectors (1 , 1) and (-1 , -1) • (d) The vectors (1 , 1) and (0 , 0)

• From (d) The vectors (1 , 1) and (0 , 0) If one of the v’s is the zero vector , independence has no chance. • The columns of A are independent exactly when the rank is r = n. There are n pivots and no free variables. Only x = 0 is in the nullspace. m • Any set of n vectors in R must be linearly dependent if n > m.

Vectors that span a subspace • Definition : A set of vectors spans a space if their linear combination fill the space.

• Definition : n The row space of a matrix. T is the subspace of R spanned by the rows. T The row space of A is C(A ). It is the column space of A. 3 T 2 • C(A) is a plan in R , C(A ) is all of R. n m • The row are in R spanning the row space. The columns are in R spanning the column space.

A Basis for a Vector Space • Definition : A basis for a vector space is a sequence of vectors that has two properties at once. (1)The vectors are linearly independent. (2)The vectors span the space. • Properties The combination that produce any vectors in the space is “unique”. There is one and only one way to write v as a combination of the basis vectors.

n • The columns of every invertible n by n matrix give a basis for R. n • R has infinitely many different basis. (There are infinitely many invertible n by n matrices. ) • Pivot columns of A are a basis for its column space. • Pivot rows of A are a basis for its row space.

• The column space of A may or may not have the same column space of R. • The row space of A is the same as the row space of R.

7 • Given 5 vectors in R , how do you find a basis for the space they span? (1)Put the vectors as the rows of A. Eliminate to find the non-zero rows of R. (2)Put the vectors as the columns of A. Eliminate to find the pivot columns of A (, not R !)

Dimension of a Vector space • All bases for a vector space contain the same number of vectors. This number is the “ dimension ” of the space. 1 2 m 1 2 n • If v , …, v and w , …, w are both bases for the same vector space , then m = n.

• The dimension of a space is the number of vectors in every basis.