linear_algebra_review_2009_09_24_01

linear_algebra_review_2009_09_24_01 - 3-1 Linear Algebra...

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3 - 1 Linear Algebra Review S. Lall, Stanford 2009.09.24.01 3. Linear Algebra Review Range and Null space Left and right invertibility Rank Conservation of dimension Invertibility
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3 - 2 Linear Algebra Review S. Lall, Stanford 2009.09.24.01 The Range The range is defned to be range( A ) = b Ax | x R n B For control problems: range( A ) = set oF possible outputs oF y = Ax = span oF columns oF A the range is also called the column space or the image oF A the range is important in control problems because The equation y = Ax has a solution x ⇐⇒ y range( A ) IF we want a solution For all y R m , then we need range( A ) = R m . We call such a matrix full-range or fully controllable or onto or surjective or right-invertible
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3 - 3 Linear Algebra Review S. Lall, Stanford 2009.09.24.01 Full Range Space range A = R m is immediately equivalent to the columns of A span R m . there is a right inverse function h : R n R m so that Ah ( y ) = y A h y y des x applied Notice that h is a controller that gives an input that exactly produces y des , since y = Ax applied = Ah ( y des ) = y des We’ll see that in fact there is a linear right inverse B such that AB = I . Not so immediately, we’ll see that range A = R m if and only if the rows of A are linearly independent. AA T is invertible.
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3 - 4 Linear Algebra Review S. Lall, Stanford 2009.09.24.01 The Null Space null( A ) = b x R n | Ax = 0 B for estimation problems: null( A ) = set of unknowns which produce zero sensor output
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linear_algebra_review_2009_09_24_01 - 3-1 Linear Algebra...

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