An Introduction To Linear Algebra - download pdf or read online

By Kenneth Kuttler

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Now let u be a vector and let t0 be some reference time. For example you could let t0 = 0. Then define the components of u with respect to these vectors, i, j, k at time t0 as u ≡u1 i (t0 ) + u2 j (t0 ) + u3 k (t0 ) . 60 MATRICES AND LINEAR TRANSFORMATIONS Let u (t) be defined as the vector which has the same components with respect to i, j, k but at time t. Thus u (t) ≡ u1 i (t) + u2 j (t) + u3 k (t) . and the vector has changed although the components have not. This is exactly the situation in the case of the apparently fixed basis vectors on the earth if u is a position vector from the given spot on the earth’s surface to a point regarded as fixed with the earth due to its keeping the same coordinates relative to the coordinate axes which are fixed with the earth.

4 Compute  1  0 2 2 2 1 1 1 4   1 3  2 −2    0 1 1   .  First of all this is of the form (3 × 4) (4 × 1) and so the result should be a (3 × 1) . Note how the inside numbers cancel. To get the entry in the second row and first and only column, compute 4 a2k vk = a21 v1 + a22 v2 + a23 v3 + a24 v4 = 0 × 1 + 2 × 2 + 1 × 0 + (−2) × 1 = 2. 1. MATRICES 37 You should do the rest of the problem and verify    1 1 2 1 3  2  0 2 1 −2    0 2 1 4 1 1    8   =  2 .  5 With this done, the next task is to multiply an m × n matrix times an n × p matrix.

In doing arithmetic with matrices you must define what happens in terms of the cij sometimes called the entries of the matrix or the components of the matrix. The above discussion stated for general matrices is given in the following definition. 1 Let A = (aij ) and B = (bij ) be two m × n matrices. Then A + B = C where C = (cij ) for cij = aij + bij . Also if x is a scalar, xA = (cij ) where cij = xaij . The number Aij will typically refer to the ij th entry of the matrix, A. The zero matrix, denoted by 0 will be the matrix consisting of all zeros.

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An Introduction To Linear Algebra by Kenneth Kuttler


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