5.5 Row Space, Column Space, and Nullspace

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• 5.5 Row Space, Column Space, and Nullspace

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Row Space, Column Space, and Nullspace

• Definition
• For an mxn matrix
• the vectors
• in Rn formed from the rows of A are called the
  row vectors of A, and the vectors
• in Rm formed from the columns of A are called
  the column vectors of A

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Row Space, Column Space, and Nullspace

• Definition
• If A is an mxn matrix, then the subspace of Rn
  spanned by the row vectors of A is called the row
  space of A,
• and the subspace of Rm spanned by the column
  vectors is called the column space of A.
• The solution space of the homogeneous system of
  equations Ax0, which is a subspace of Rn, is
  called the nullspace of A.
• Theorem 5.5.1 A system of linar equations Axb
  is consistent iff b is in the column space of A

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example
Show that b in column space of A! The solution by
G.E. X1 2, X2 -1, X3 3, the system is
consistent, b is in the column space of A
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Row Space, Column Space, and Nullspace

• Theorem 5.5.2 If x0 denotes any single solution
  of a consistent linear system Axb, and if
  v1,v2,...,vk form a basis for the nullspace of A,
  that is, the solution space of the homogeneous
  system Ax0, then every solution of Axb can be
  expressed in the form
• and, conversely, for all choices of scalars
  c1,c2,...,ck, the vector x in this formula is a
  solution of Axb.

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General and Particular Solutions

• Terminology
• Vector x0 is called a particularly solution of
  Axb.
• The expression x0c1v1c2v2...ckvk is called
  the general solution of Axb.
• The expression c1v1c2v2...ckvk is called the
  general solution of Ax0.

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Bases for Row Spaces, Column Spaces, and
Nullspaces

• Theorem 5.5.3 Elementary row operations do not
  change the nullspace of a matrix
• Theorem 5.5.4 Elementary row operations do not
  change the row space of a matrix
• Theorem 5.5.5 If A and B are row equivalent
  matrices, then
• A given set of column vectors of A is linearly
  independent iff the corresponding column vectors
  of B are linearly independent.
• A given set of column vectors of A forms a basis
  for the column space of A iff the corresponding
  column vectors of B form a basis for the column
  space of B.

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Bases for Row Spaces, Column Spaces, and
Nullspaces

• Theorem If a matrix R is in row-echelon form,
  then the row vectors with the leading 1s (i.e.,
  the nonzero row vectors) form a basis for the row
  space of R, and the column vectors with the
  leading 1s of the row vectors form a basis for
  the column space of R.
• Example Bases for Row and Column Spaces
• The matrix R is in row-echelon form, while the
  vectors r
• form a basis for the row space of R

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Bases for Row Spaces, Column Spaces, and
Nullspaces

• and the vectors
• form a basis for the column space of R
• Example Bases for Row and Column Spaces
• Find bases for the row and column spaces of

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Bases for Row Spaces, Column Spaces, and
Nullspaces

• The basis vectors are
• The first, third, and fifth columns of R contain
  the leading 1s of the row vectors that form a
  basis for the column space of R.
• Thus the corresponding column vectors of A, form
  a basis for the column space of A

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Bases for Row Spaces, Column Spaces, and
Nullspaces

• Example Basis and Linear Combinations
• Find a subset of the vectors
• v1(1,-2,0,3), v2(2,-5,-3,6), v3(0,1,3,0),
• v4(2,-1,4,-7), v5(5,-8,1,2) that forms a basis
  for the space spanned by these vectors.
• Express each vector not in the basis as a linear
  combination of the basis vector
• Solution

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Bases for Row Spaces, Column Spaces, and
Nullspaces

• Basis for the column space of matrix vectors w
  is w1,w2,w4 and consequently basis for the
  column space of matrix vectors v is v1,v2,v4.
• Expressing w3 and w5 as linear combinations of
  the basis vectors w1,w2, and w4 (dependency
  equations).
• w3 2w1 - w2
• w5 w1 w2 w4
• The corresponding relationships are
• v3 2v1 v2
• v5 v1 v2 v4

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Bases for Row Spaces, Column Spaces, and
Nullspaces

• Given a set of vectors Sv1,v2,...,vk) in Rn,
  the following procedure produces a subset of
  these vectors that forms a basis for span(S) and
  expresses those vectors of S that are not in the
  basis as linear combinations of the basis
  vectors.
• Step 1. Form the matrix A having v1,v2,...,vk as
  its column vectors.
• Step 2. Reduce the matrix A to its reduced
  row-echelon form R, and let w1,w2,...,wk be the
  column vectors of R.
• Step 3. Identify the columns that contain the
  leading 1s in R. The corresponding column
  vectors of A are the basis vectors for span(S).
• Step 4. Express each column vector of R that
  does not contain a leading 1 as a linear
  combination of preceding column vectors that do
  contain leading 1s.

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• 5.6 Rank and Nullity

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Four Fundamental Matrix Spaces

• Fundamental matrix spaces
• Row space of A, Column space of A
• Nullspace of A, Nullspace of AT
• Relationships between the dimensions of these
  four vector spaces.

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Row and Column Spaces have Equal Dimensions

• Theorem 5.6.1 If A is any matrix, then the row
  space and column space of A have the same
  dimension.
• The common dimension of the row space and column
  space of a matrix A is called the rank of A and
  is denoted by rank(A) the dimension of the
  nullspace of A is called the nullity of A and is
  denoted by nullity(A).

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Row and Column Spaces have Equal Dimensions

• Example Rank and Nullity of a 4x6 Matrix
• Find the rank and nullity
• of the matrix
• Solution
• The reduced row-echeclon
• form of A is
• rank(A) 2 and the corresponding system will be

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Row and Column Spaces Have Equal Dimensions

• The general solution of the system is

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Row and Column Spaces Have Equal Dimensions

• Nullity(A)4

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Row and Column Spaces Have Equal Dimensions

• Theorem 5.6.2 If A is any matrix, then rank(A)
  rank(AT).
• Theorem 5.6.3 Dimension Theorem for Matrices
• If A is a matrix with n columns, then
• rank(A) nullity(A) n
• Theorem 5.6.4 If A is an mxn matrix, then
• Rank(A) the number of leading variables in the
  solution of Ax 0.
• Nullity(A) the number of parameters in the
  general solution of Ax 0.

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Row and Column Spaces Have Equal Dimensions

• A is an mxn matrix of rank r

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Maximum Value for Rank

• A is an mxn matrix
• rank(A) min(m,n)
• where min(m,n) denotes the smaller of the
  numbers m and n if m?n or their common value if
  mn.

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Linear Systems of m Equations in n Unknowns

• Theorem 5.6.5 The Consistency Theorem
• If Ax b is a linear system of m equations in n
  unknowns, then the following are equivalent.
• Ax b is consistent
• b is in the column space of A.
• The coefficient matrix A and the augmented matrix
  Ab have the same rank.

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Linear Systems of m Equations in n Unknowns

• Theorem
• If Ax b is a linear system of m equations in n
  unknowns, then the following are equivalent.
• Ax b is consistent for every mx1 matrix b.
• The column vectors of A span Rm.
• Rank(A) m
• A linear system with more equations than unknowns
  is called an overdetermined linear system. The
  system cannot be consistent for every possible b.

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Linear Systems of m Equations in n Unknowns

• Example Overdetermined System
• The system is consistent
• iff b1, b2, b3, b4, and b5
• satisfy the conditions

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Linear Systems of m Equations in n Unknowns

• Theorem 5.6.7 If Axb is a consistent linear
  system of m equations in n unknowns, and if A has
  rank r, then the general solution of the system
  contains n-r parameters.
• Theorem 5.6.8 If A is an mxn matrix, then the
  following are equivalent.
• Ax0 has only the trivial solution.
• The column vectors of A are linearly independent.
• Axb has at most one solution (none or one) for
  every mx1 matrix b.
• A linear system with more unknowns than equations
  is called an underdetermined linear system.
• Underdetermined linear system is consistent if
  its solution has at least one parameter ? has
  infinitely many solution.

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Summary

• Theorem 5.6.9 Equivalent Statements
• If A is an nxn matrix, and if TARn?Rn is
  multiplication by A, then the following are
  equivalent.
• A is invertible
• Ax0 has only the trivial solution
• The reduced row-echelon form of A is In.
• A is expressible as a product of elementary
  matrices.
• Axb is consistent for every nx1 matrix b
• Axb has exactly one solution for every nx1
  matrix b
• Det(A)?0

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Summary

• The range of TA is Rn
• TA is one-to-one
• The column vectors of A are linearly independent
• The row vectors of A are linearly independent
• The column vectors of A span Rn
• The row vectors of A span Rn
• The column vectors of A form a basis for Rn
• The row vectors of A form a basis for Rn
• A has rank n
• A has nullity 0