vector_spaces4

vector_spaces4 - ROW SPACE, COLUMN SPACE, and NULLSPACE...

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1 ROW SPACE, COLUMN SPACE, and NULLSPACE EXAMPLE List the Row and Column Vectors in 3 ä 4 Matrix 12 3 3 4 21 0 1 35 71 14 27 The row vectors are (2 1 0 1), (3 5 7 1), (1 4 2 7) Thecolumn vectors are 0 1 3, 5, 7, 1 2 7 Solution A ⎛⎞ ⎜⎟ =− ⎝⎠ = = ⎛ ⎞ ⎜ ⎟ == = = ⎝ ⎠ rr r cc c c DEFINITION If A is an m x n matrix, then the subspace of R n spanned by the row vectors of A is called the row space of A, and the subspace of R m spanned by the column vectors of A is called the column space of A. The solution space of the homogeneous system of equations A x = 0 , which is a subspace of R n , is called the nullspace of A. _
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2 THEOREM 5.5.1 A system of linear equations A x = b is consistent if and only if b is in the column space of A. The next theorem establishes a fundamental relationship between the solutions of a nonhomogeneous linear system A x = b and those of the corresponding homogeneous linear system A x = 0 with the same coefficient matrix. THEOREM 5.5.2 If x 0 denotes any single solution of a consistent linear system A x = b and v 1, v 2 ,…, v k form a basis for the
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vector_spaces4 - ROW SPACE, COLUMN SPACE, and NULLSPACE...

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