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vector_spaces5

# vector_spaces5 - RANK and NULLITY 1 Row and Column Spaces...

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1 RANK and NULLITY 1. 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 dim(row space of A ) = dim(column space of A ) DEFINITION 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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2 2. EXAMPLE 1 Find the rank and nullity of the 4 x 6 matrix -1 2 0 4 5 -3 3- 7 201 4 2- 5 24 6 1 4 -9 2 -4 -4 7 A ⎛⎞ ⎜⎟ = ⎝⎠ Solution The RREF of A is 1 0 -4 -28 -37 13 0 1 -2 -12 -16 5 00 0 0 0 0 A = There are two nonzero rows (or, equivalently, two leading l's), the row space and column space are both two-dimensional, so rank( A ) = 2. To find the nullity of A , we must find the dimension of the solution space of the linear system A x = 0 .
3 Solving for the leading variables x 1 4 x 3 – 28 x 4 – 37 x 5 + 13 x 6 = 0 x 2 – 2 x 3 – 12 x 4 – 16 x 5 + 5 x 6 = 0 we get x 1 = 4 x 3 + 28 x 4 + 37 x 5 – 13 x 6 x 2 = 2 x 3 + 12 x 4 + 16 x 5 – 5 x 6 The general solution of the system is x 1 = 4 r + 28 s + 37 t – 13 u x 2 = 2 r + 12 s + 16 t – 5u x 3 = r x 4 = s x 5 = t x 6 = u or, equivalently, 1 2 3 4 5 6 4r + 28s + 37t - 13u 4 28 37 2r + 12s + 16t - 5u 2 12 16 r1 0 0 s 0 1 0 t 0 0 1 u 0 0 0 x x x rs t x x x ⎛⎞ ⎜⎟ == + + ⎝⎠ 13 5 0 0 0 1 u

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vector_spaces5 - RANK and NULLITY 1 Row and Column Spaces...

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