4 © 2012 Pearson Education, Inc. Vector Spaces 4.4 COORDINATE SYSTEMS

Slide © 2012 Pearson Education, Inc. THE UNIQUE REPRESENTATION THEOREM  Theorem 7: Let B be a basis for vector space V. Then for each x in V, there exists a unique set of scalars c 1, …, c n such that ----(1)

Slide © 2012 Pearson Education, Inc. THE UNIQUE REPRESENTATION THEOREM  Definition: Suppose B is a basis for V and x is in V. The coordinates of x relative to the basis B (or the B -coordinate of x) are the weights c 1, …, c n such that.

Slide © 2012 Pearson Education, Inc. THE UNIQUE REPRESENTATION THEOREM  If c 1, …, c n are the B -coordinates of x, then the vector in [x] B is the coordinate vector of x (relative to B ), or the B -coordinate vector of x.  The mapping B is the coordinate mapping (determined by B ).

Slide © 2012 Pearson Education, Inc. COORDINATES IN  When a basis B for is fixed, the B -coordinate vector of a specified x is easily found, as in the example below.  Example 1: Let,,, and B. Find the coordinate vector [x] B of x relative to B.  Solution: The B -coordinate c 1, c 2 of x satisfy b1b1 b2b2 x

Slide © 2012 Pearson Education, Inc. COORDINATES IN or ----(3)  This equation can be solved by row operations on an augmented matrix or by using the inverse of the matrix on the left.  In any case, the solution is,.  Thus and. b1b1 b2b2 x

Slide © 2012 Pearson Education, Inc. COORDINATES IN  See the following figure.  The matrix in (3) changes the B -coordinates of a vector x into the standard coordinates for x.  An analogous change of coordinates can be carried out in for a basis B.  Let P B

Slide © 2012 Pearson Education, Inc. COORDINATES IN  Then the vector equation is equivalent to ----(4)  P B is called the change-of-coordinates matrix from B to the standard basis in.  Left-multiplication by P B transforms the coordinate vector [x] B into x.  Since the columns of P B form a basis for, P B is invertible (by the Invertible Matrix Theorem).

Slide © 2012 Pearson Education, Inc. COORDINATES IN  Left-multiplication by converts x into its B - coordinate vector:  The correspondence B, produced by, is the coordinate mapping.  Since is an invertible matrix, the coordinate mapping is a one-to-one linear transformation from onto, by the Invertible Matrix Theorem.

Slide © 2012 Pearson Education, Inc. THE COORDINATE MAPPING  Theorem 8: Let B be a basis for a vector space V. Then the coordinate mapping B is a one-to-one linear transformation from V onto.

Slide © 2012 Pearson Education, Inc. THE COORDINATE MAPPING  Every vector space calculation in V is accurately reproduced in W, and vice versa.  In particular, any real vector space with a basis of n vectors is indistinguishable from.  Example 2: Let,,, and B. Then B is a basis for. Determine if x is in H, and if it is, find the coordinate vector of x relative to B.

Slide © 2012 Pearson Education, Inc. THE COORDINATE MAPPING  Solution: If x is in H, then the following vector equation is consistent:  The scalars c 1 and c 2, if they exist, are the B - coordinates of x.

Slide © 2012 Pearson Education, Inc. THE COORDINATE MAPPING  Using row operations, we obtain.  Thus, and [x] B.

Slide © 2012 Pearson Education, Inc. THE COORDINATE MAPPING  The coordinate system on H determined by B is shown in the following figure.

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4 Vector Spaces 4.5 THE DIMENSION OF A VECTOR SPACE

Slide © 2012 Pearson Education, Inc. DIMENSION OF A VECTOR SPACE  Theorem 9: If a vector space V has a basis B, then any set in V containing more than n vectors must be linearly dependent.  Theorem 9 implies that if a vector space V has a basis B, then each linearly independent set in V has no more than n vectors.

Slide © 2012 Pearson Education, Inc. DIMENSION OF A VECTOR SPACE  Definition: If V is spanned by a finite set, then V is said to be finite-dimensional, and the dimension of V, written as dim V, is the number of vectors in a basis for V. The dimension of the zero vector space {0} is defined to be zero. If V is not spanned by a finite set, then V is said to be infinite-dimensional.  Example 1: Find the dimension of the subspace

Slide © 2012 Pearson Education, Inc. DIMENSION OF A VECTOR SPACE  H is the set of all linear combinations of the vectors,,,  Clearly,, v 2 is not a multiple of v 1, but v 3 is a multiple of v 2.  By the Spanning Set Theorem, we may discard v 3 and still have a set that spans H.

Slide © 2012 Pearson Education, Inc. SUBSPACES OF A FINITE-DIMENSIONAL SPACE  Finally, v 4 is not a linear combination of v 1 and v 2.  So {v 1, v 2, v 4 } is linearly independent and hence is a basis for H.  Thus dim.  Theorem 11: Let H be a subspace of a finite- dimensional vector space V. Any linearly independent set in H can be expanded, if necessary, to a basis for H. Also, H is finite-dimensional and

Slide © 2012 Pearson Education, Inc. THE BASIS THEOREM  Theorem 12: Let V be a p-dimensional vector space,. Any linearly independent set of exactly p elements in V is automatically a basis for V. Any set of exactly p elements that spans V is automatically a basis for V.

Slide © 2012 Pearson Education, Inc. DIMENSIONS OF NUL A AND COL A  Thus, the dimension of Nul A is the number of free variables in the equation, and the dimension of Col A is the number of pivot columns in A.  Example 2: Find the dimensions of the null space and the column space of

Slide © 2012 Pearson Education, Inc. DIMENSIONS OF NUL A AND COL A  Solution: Row reduce the augmented matrix to echelon form:  There are three free variable—x 2, x 4 and x 5.  Hence the dimension of Nul A is 3.  Also dim Col because A has two pivot columns.

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4 Vector Spaces 4.6 RANK

Slide © 2012 Pearson Education, Inc. THE ROW SPACE  If A is an matrix, each row of A has n entries and thus can be identified with a vector in.  The set of all linear combinations of the row vectors is called the row space of A and is denoted by Row A.  Each row has n entries, so Row A is a subspace of.  Since the rows of A are identified with the columns of A T, we could also write Col A T in place of Row A.

Slide © 2012 Pearson Education, Inc. THE ROW SPACE  Theorem 13: If two matrices A and B are row equivalent, then their row spaces are the same. If B is in echelon form, the nonzero rows of B form a basis for the row space of A as well as for that of B.

Slide © 2012 Pearson Education, Inc. THE ROW SPACE  Example 1: Find bases for the row space, the column space, and the null space of the matrix  Solution: To find bases for the row space and the column space, row reduce A to an echelon form:

Slide © 2012 Pearson Education, Inc. THE ROW SPACE  By Theorem 13, the first three rows of B form a basis for the row space of A (as well as for the row space of B).  Thus Basis for Row

Slide © 2012 Pearson Education, Inc. THE ROW SPACE  For the column space, observe from B that the pivots are in columns 1, 2, and 4.  Hence columns 1, 2, and 4 of A (not B) form a basis for Col A: Basis for Col  Notice that any echelon form of A provides (in its nonzero rows) a basis for Row A and also identifies the pivot columns of A for Col A.

Slide © 2012 Pearson Education, Inc. THE ROW SPACE  However, for Nul A, we need the reduced echelon form.  Further row operations on B yield

Slide © 2012 Pearson Education, Inc. THE ROW SPACE  The equation is equivalent to, that is,  So,,, with x 3 and x 5 free variables.

Slide © 2012 Pearson Education, Inc. THE ROW SPACE  The calculations show that Basis for Nul  Observe that, unlike the basis for Col A, the bases for Row A and Nul A have no simple connection with the entries in A itself.

Slide © 2012 Pearson Education, Inc. THE RANK THEOREM  Definition: The rank of A is the dimension of the column space of A.  Since Row A is the same as Col A T, the dimension of the row space of A is the rank of A T.  The dimension of the null space is sometimes called the nullity of A.  Theorem 14: The dimensions of the column space and the row space of an matrix A are equal. This common dimension, the rank of A, also equals the number of pivot positions in A and satisfies the equation

Slide © 2012 Pearson Education, Inc. THE RANK THEOREM

Slide © 2012 Pearson Education, Inc. THE RANK THEOREM  Example 2: a.If A is a matrix with a two-dimensional null space, what is the rank of A? b.Could a matrix have a two-dimensional null space?  Solution: a.Since A has 9 columns,, and hence rank. b.No. If a matrix, call it B, has a two- dimensional null space, it would have to have rank 7, by the Rank Theorem.

Slide © 2012 Pearson Education, Inc. THE INVERTIBLE MATRIX THEOREM (CONTINUED)  But the columns of B are vectors in, and so the dimension of Col B cannot exceed 6; that is, rank B cannot exceed 6.  Theorem: Let A be an matrix. Then the following statements are each equivalent to the statement that A is an invertible matrix. m.The columns of A form a basis of. n.Col o.Dim Col p.rank

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