Section 5.1 - Theory, o f SysteDls o f Linear Equations and

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Theory, of SysteDls Linear Equations and Eigenvalue /Eigenvector Problel1ls Null Space and Range a Matrix The null space Null(A) of a matrix A is the set of vectors x that are solutions to the system of equations Ax O. The range Range(A) of A is the set of vectors b such that Ax = b has a solution. Another name that is sometimes used for Null(A) is the kernel of A. In this section we examine these two important sets of vectors associated with any matrix. We look at their role in linear models and learn how to determine these sets. Both the Null(A) and Range(A) are vector spaces. Definition. A vector space is any set V of vectors such that if Xl> X 2 are in V, then any linear combination rxl + SX 2 is also in V. IfAxl -= 0 and AX2 = 0, then we have A(rxl + SX 2 ) = r(Ax l ) + s(Ax 2 ) = reO) + s(O) = 0 Thus Null(A) is a vector space. A similarly simple proof, left as an exercise, shows that Range(A) is a vector space. Suppose that A is an n-by-n matrix for which the system Ax = b has a unique solution for every b. Then Range(A) is all possible n-vectors, and Null(A) is just the zero vector 0, since 0 is always a solution to Ax = 0 and by assumption there can only be one solution to Ax = O. 393
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394 Ch. 5 Theory of Systems of Linear Equations and EigenvaluelEigenvector Problems y y --+-~--~----------~x Figure 5.1 (a) Two lines intersect at a point. (b) Two parallel lines have no inter- section points. In this section we are concerned with matrices A for which Ax = b has nonunique solutions or no solutions. Then determining the range and null space of A becomes important. Let us briefly describe the geometry of systems of equations that do not have unique solutions. Consider the pair of equations 2x - Y 5 (1) 3x + 2y II In Figure 5.la we have plotted the graph of the two equations in (1) in x - y space. The graph of a linear equation is a straight line. An (x, y) pair that solves (1) must lie on the lines of both equations. That is, the (x, y) pair must be the coordinates of the point where the two lines intersect. From Figure 5 .Ia we see that this intersection point has coordinates (3, 1). Suppose the two equations produce lines that are parallel: y 4x 5 4 (2) (see Figure 5.1b). Then there is no common point--no solution to (2). Note that "parallel" means that the second equation's coefficients are multiples of the first's. We saw in the canoe-with-sail example in Section 1.1 that when two equations produce almost parallel lines, the equations can give strange results. A system of three equations in three unknowns, such as + 5y + 4z 4 x + 4y + 3z = 1 - x + 3y + 2z = - 5 (3)
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Sec. 5.1 Null Space and Range of a Matrix -;~------------~~--+---.x (a) Figure 5.2 (a) Three planes intersect at a point. (b) The lines formed by in- tersections of two planes are parallel. (c) Three planes intersect along a line. 395 Intersection lines of two planes (b) y ~~------~--------------x (e) has a simihrr interpretation in three-dimensional space (see Figure S.2a).
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Section 5.1 - Theory, o f SysteDls o f Linear Equations and

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