Chapter 4.7 Flashcards
Row vectors and column vectors
In an mxn matrix A, the m rows create the m row vectors and the n columns create the n column vectors
Row space, column space and null space
In an mxn matrix A, the subspace R^(n) spanned by the row vectors is called the row space of A and the subspace R^(m) spanned by the column vectors of A is called the coulmn space of A. The solution space of the homogeneous system Ax=0 is called the null space of A
What are the two general questions of chapter 4.7
1) What relationship exist among between the solutions of a linear system Ax=b and a row space, column space and null space of the coefficient matrix A.
2) What relationship exist among the row space, column space and null space of a matrix.
If x0 is any solution of a constant linear system Ax=0, and if S={v1, v2, … , vk} is a basis for the null space of A then
Every solution of Ax=b can be expressed in the form
x = x0 + c1v1 + c2v2 + … + ckvk
conversely, for all choices of scalars c1, c2, … , ck, the vector x in this formula is a solution of Ax=b
A system of linear equations Ax=b is consistemt iff
b is in the column space of A
The general solution of a consistemt linear system can be expressed as
The sum of a particular solution of that system and a general solution of the corresponding homogeneous system.
What relation is there between elementary row operations and the row space, column space and null space of a matrix?
Row operations do not change the row space or null space of a matrix but may change the column space
Find bases for the row and column spaces of a matrix in row echelon form by inspection
If a matrix R is in row echelon form, then the row vectors with leading 1’s (the nonzero row vectors) form a basis for the row space of A and the column vectors with leading 1’s form a basis for the column space of R
If A and B are row equivalent matrices then (2)
1) A given set of column vectors of A is linearly independent iff the corresponding column vectors of B are linearly independent.
2) A given set of column vectors of Aform a basis for the column space of A iff the corresponding column vectors of B form a basis for the column space of B.
The Basis problem of chapter 4.7
Given a set of vectors S={v1, v2, … , vk} in R^(n), find a subset of these vectors that form a basis for span(s), and express those vectors that are not in that basis as a linear combination of the basis vectors.
Basis for Span(S) (4+2 step guide)
1) Form the matrix A having vectors in S={v1, v2, … , vk} as column vectors
2) Reduce the matrix A to reduced row echelon form R.
3) Denote the column vectors of R by w1, w2, … , wk
4) Identify the columns of R that contain the leading 1’s. The corresponding column vectors of A form a basis for span(S)
5) Obtain a set of dependency equations by expressing each column vector of R that does not contain a leading 1 as a linear combination of preceding column vectors that do countain leading 1’s.
6) Replace the column vectors of R that apear in the dependency equations by the corresponding column vectors of A.
