Linear Algebra. A Geometric Approach by Ted Shifrin, Malcolm Adams

By Ted Shifrin, Malcolm Adams

Linear Algebra: a geometrical procedure, moment Edition, provides the normal computational facets of linear algebra and incorporates a number of fascinating fascinating purposes that may be fascinating to inspire technological know-how and engineering scholars, in addition to aid arithmetic scholars make the transition to extra summary complex classes. The textual content publications scholars on easy methods to take into consideration mathematical techniques and write rigorous mathematical arguments.

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Of course, planes can be described in different ways. 14 Chapter 1 Vectors and Matrices EXAMPLE 9 We wish to find a parametric equation of the plane that contains the points P = (1, 2, 1) −→ and Q = (2, 4, 0) and is parallel to the vector (1, 1, 3). We take x0 = (1, 2, 1), u = P Q = (1, 2, −1), and v = (1, 1, 3), so the plane consists of all points of the form x = (1, 2, 1) + s(1, 2, −1) + t (1, 1, 3), s, t ∈ R. Finally, note that three noncollinear points P , Q, R ∈ R3 determine a plane. To get a −→ −→ −→ parametric equation of this plane, we simply take x0 = OP , u = P Q, and v = P R.

If c1 , . . , ck ∈ R, the vector v = c1 v1 + c2 v2 + · · · + ck vk is called a linear combination of v1 , . . , vk . 15 12 Chapter 1 Vectors and Matrices Definition. Let v1 , . . , vk ∈ Rn . The set of all linear combinations of v1 , . . , vk is called their span, denoted Span (v1 , . . , vk ). That is, Span (v1 , . . , vk ) = {v ∈ Rn : v = c1 v1 + c2 v2 + · · · + ck vk for some scalars c1 , . . , ck }. In terms of our new language, then, the span of two nonparallel vectors u, v ∈ Rn is a plane through the origin.

16 Chapter 1 Vectors and Matrices 6. Find a parametric equation of each of the following lines: a. 3x1 + 4x2 = 6 ∗ b. the line with slope 1/3 that passes through A = (−1, 2) c. the line with slope 2/5 that passes through A = (3, 1) d. the line through A = (−2, 1) parallel to x = (1, 4) + t (3, 5) e. the line through A = (−2, 1) perpendicular to x = (1, 4) + t (3, 5) ∗ f. the line through A = (1, 2, 1) and B = (2, 1, 0) g. the line through A = (1, −2, 1) and B = (2, 1, −1) ∗ h. the line through (1, 1, 0, −1) parallel to x = (2 + t, 1 − 2t, 3t, 4 − t) 7.

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