Basic Linear Algebra, Edition: version 8 Dec 2009 by Andrew Baker

By Andrew Baker

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Extra info for Basic Linear Algebra, Edition: version 8 Dec 2009

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A) Write down the matrix of f with respect to the standard bases of R3 and R2 . (b) Determine the matrix T [f ]S , where S and T are the bases S : (1, 0, 1), (0, 1, 1), (0, 0, 1), T : (3, 1), (2, 1). Solution. (a) In terms of the standard basis vectors (1, 0, 0), (0, 1, 0), (0, 0, 1) and (1, 0), (0, 1) we have f (1, 0, 0) = (2, 1) = 2(1, 0) + 1(0, 1), f (0, 1, 0) = (−3, 0) = −3(1, 0) + 0(0, 1), f (0, 0, 1) = (7, −1) = 7(1, 0) + (−1)(0, 1), hence the matrix is 2 −3 7 . 4. APPLICATION TO MATRICES AND SYSTEMS OF LINEAR EQUATIONS so we take 41   1 0 0   A = 0 1 0 .

5. Geometric linear transformations When considering a linear transformation f : Rn −→ Rn on the real vector space Rn it is often important or useful to understand its geometric content. 43. For θ ∈ R, consider the linear transformation ρθ : R2 −→ R2 given by ρθ (x, y) = (x cos θ − y sin θ, x sin θ + y cos θ). Investigate the geometric effect of ρθ on the plane. Solution. The effect on the standard basis vectors is ρθ (1, 0) = (cos θ, sin θ), ρθ (0, 1) = (− sin θ, cos θ). Using the dot product of vectors it is possible to check that the angle between ρθ (x, y) and (x, y) is always θ if (x, y) = (0, 0), and that the lengths of ρθ (x, y) and (x, y) are equal.

A) Let x ∈ V have coordinates x1 , . . , xn with respect to the basis S : v1 , . . , vn of V and let f (x) ∈ W have coordinates y1 , . . , ym with respect to the basis T : w1 , . . , wm of W . Then        x1 t11 · · · t1n x1 y1  .   .  .  .  . .. . 3a)  . .  .  = T [f ]S  .  =  . 3b) T [f ]S = B −1 T [f ]S A. Proof. These follow from calculations with the above formulae. 3), allow us to reduce calculations to ones with coordinates and matrices. 36. Consider the linear transformation f : R3 −→ R2 ; f (x, y, z) = (2x − 3y + 7z, x − z) between the real vector spaces R3 and R2 .

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