Term
row space of A
(A is an mxn matrix) |
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Definition
| the subspace of Rn spanned by the row vectors of A. notated as row(A) |
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Term
column space of A
(let A be an mxn matrix) |
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Definition
| the subspace of Rm spanned by the column vectors of A. notated as col(A) |
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row-equivalent matrices have the same row space
(theorem 4.13) |
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Definition
| if an mxn matrix A is row-equivalent to an mxn matrix, then row(A)=row(B) |
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Term
basis for the row space of a matrix
(theorem 4.14) |
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Definition
| if a matrix A is row-equivalent to a matrix B in row echelon form, then the nonzero row vecetors of B form a basis for row(A) |
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Term
| row and column spaces have equal dimensions |
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Definition
| if A is an mxn matrix, then the row space and column space of A have the same dimension |
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Term
rank of A:
denoted by rank(A) |
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Definition
| the dimension of the row (or column) space of a matrix |
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solutions of a homogeneous system
(theorem 4.16) |
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Definition
if A is an mxn matrix, then the set of all solutions of the homogeneous system of linear equations Ax=0 is a subspace of Rn called the null space of A, denoted by N(A)
N(A)={xE Rn:Ax=0} |
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Term
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Definition
| dimension of the nullspace of A |
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Term
dimension of the solution space
(theorem 4.17) |
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Definition
if A is an mxn matrix of rank r, then the dimension of the solution of Ax=0 is n-r. that is
n=rank(A)+nullity(A)
n=r+nullity(A)
n-r=nullity(A) |
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Term
solutions of a nonhomogeneous linear system
(theorem 4.18) |
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Definition
| if xp is a fixed (particular) solution of a nonhomogeneous system Ax=b, then every solution of this system can be written of the form x=xp+xh where xh is a solution of the corresponding homogeneous system Ax=0 |
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Term
solutions of a system of linear equations
(theorem 4.19) |
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Definition
| the system Ax=b is consistent if and only if b is in the column space of A |
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equivalent conditions
(aka the invertible matrix theorem) |
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Definition
if A is an invertible matrix, then the following conditions are equivalent
1) A is invertible
2) Ax=b has a unique solution for an nx1 matrix b(spanning)
3) Ax=0 has only the trivial solution (LI)
4) A is row-equivalent to In
5) |A|≠0
6) rank(A)=n
7) the n row vectors of A are linearly independent
8) the n column vectors of A are linearly independent |
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