Section 5.2 - Sec 5.2 Theory of Vector Spaces Associated...

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Sec. 5.2 Theory of Vector Spaces Associated with Systems of Equations 413 23. Give a constraint equation, if one exists, on the probability vectors in the range of the transition matrices in Exercise 15. 24. By looking at the transition matrix for the frog Markov chain, explain why the even-state probabilities in the next period must equal the odd- state probabilities in the next period. Hint: Show that this equality is true if we start (this period) from a specific state. 25. Compute the constraint on the range of the following powers of the frog transition matrix. You will need a matrix software package. (a) A2 (b) A3 (c) A!O 26. Show that the range R(A) of a matrix is a vector space. That is, if b and b' are in (for some x and x', Ax = b and Ax' = b'), show that rb + sb' is in R(A), for any scalars r, s. 27. Using matrix algebra, show that if x! and x 2 are solutions to the matrix equation Ax b, then any linear combination x' = ex! + dx 2 , with e + d = . 1, is also a solution. 28. Show that the intersection VI n V 2 of two vector spaces V!, V 2 is again a vector space. Theory of Vector Spaces Associated with Systems Equations In this section we introduce basic concepts abouJ vector spaces and use them to obtain important information about the rangeand null space of a matrix. Recall that a vector space V is a collection of vectors such that if u, v E V, then any linear combination ru + sv is in V. In Section 5.1 we introduced the range and null space of a matrix A: Range(A) = {b : Ax Null(A) = {x : Ax b for some x} O} We noted that Range(A) and Null(A) are both vector spaces. In Examples 2, 3, and 4 of Section 5.1 we used the elimination process to find a vector or pair of vectors that generated the null spaces of certain matrices. For example, multiples of [ -1, 2, - 2, 2, - 2, 1] formed the null space of the frog Markov transition matrix. In Examples 7, 8, and 9 of Section 5.1, we used elimination to find constraint equations that vectors in the range must satisfy. For the frog Markov matrix, the constraint for range vectors p was PI + P3 + Ps = P2 + P4 + P6'
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414 Ch. 5 Theory of Systems of Linear Equations and EigenvaluelEigenvector Problems The number of vectors generating the null space and number of con- straint equations for the range were dependent on how many pivots we made during elimination. Our goal in this section is to show that the sizes of Null(A) and Range(A) are independent of how elimination is performed. The vector space V generated by a set Q = {q!, q2, ... , q,} of vectors is the collection of all vectors that can be expressed as a linear combination of the q;'s. That is, V = {v: v For example, if Q consists of the unit n-vectors e j (with all O's except for a 1 in positionj), then V is the vector space of all n-vectors, that is, euclidean n-space. Another name for a generating set is a spanning set. The column space of A, denoted Col (A), is the vector space generated by the column vectors af of A. When we write Ax = b as we see that the system Ax b has a solution if and only if b can be expressed as a linear combination of the column vectors of A, or Lemma 1. The system Ax =
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Section 5.2 - Sec 5.2 Theory of Vector Spaces Associated...

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