By Jorge Rebaza
ISBN10: 1118229622
ISBN13: 9781118229620

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Then, dimcol(A) + dimAT(^l) = n. 58 A = ri 4 7i 2 5 8 . Observe that we can write the third column a^ as a combi_3 6 9J nation of the first two oi, 0,2 as as = 2a,2 — a\. And since these first two columns are linearly independent, this implies that co\(A) is 2dimensional. Geometrically then, col(^4) is the plane — x + 2y — z = 0 passing through the origin, with normal vector [—1 2 — 1] T . 48, the null space N(A) is onedimensional. In fact, by solving the system Ax = 0, we observe that all solutions are of the form [xi X2 X3]7\ with X2 = — 2#i, xs = x\.
Then, this equality gives the system 2ci + c2 c\ + 3c2 = = 0 0, whose solution is ci = c2 = 0. Therefore, vi and V2 are linearly independent. Observe that in this case, linear independence means the vectors are not parallel or. multiple of each other. 4. 36 is not linearly independent. In fact, any of the vectors in Si can be written as a combination of the other two. 37 is linearly independent. 41 Let V = P2 be the vector space of real polynomials of degree at most 2. The set { /1 = x2 + Sx  1, / 2 = x + 3, / 3 = 2x2  x + 1} is linearly independent.
In general, due to the way they are defined, permutation matrices satisfy the important property PTP = 7 = PPT. 25) Therefore, they are invertible, and P l = PT. Multiplying an arbitrary matrix A by a permutation matrix P from the left will result in row permutations in A, and multiplying from the right will result in column permutations. 18). Then, we can simultaneously interchange rows and columns of a given matrix A through the operation PT A P. In the example below, AP is the matrix A with the first and third columns interchanged.
A First Course in Applied Mathematics by Jorge Rebaza
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