What you can now do.
Six ideas, all built on one operation: combining columns.
- §1.1 Linear combinations. Scale and add: steps of v and steps of w.
- §1.3 Ax by columns. The basket: each column is a food, x is how many of each, Ax is the total. Compute by rows, think by columns.
- §1.3, §3.1 Column space. Everywhere the boat can reach. Ax = b is solvable exactly when b is in C(A).
- §2.1 Row and column pictures. Two bearings crossing, or a recipe that makes b. Same answer.
- §8.1, §5.3 Transformations. Graph paper pinned at the origin. The columns are where the two landmarks land, and |det A| is the area factor.
- §2.4 Products. Socks, then shoes. AB applies B first.
Next lesson: elimination (§2.2, §2.3, §2.6)
Solve Ax = b the way a computer does. Subtract multiples of one row from another until the matrix is triangular. Then solve from the bottom up. Write those steps as matrices and you get A = LU.
Read along
Gilbert Strang, Introduction to Linear Algebra, 5th edition (2016), chapter 1 and §2.1. His MIT 18.06 lectures are free on MIT OpenCourseWare. For more pictures, watch 3Blue1Brown's Essence of Linear Algebra.
Review problems
Try each one before you open the solution.
A = 1326. Describe C(A). Is (2, 4) in it?
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Column 2 = 3·column 1, so C(A) is the line through (1, 2). (2, 4) = 2·(1, 2) is on it, for example x = (2, 0).
Build the matrix that turns the plane by 90° and doubles every length. What is its determinant?
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2R = 0−220. Lengths double, so areas grow 4 times: det = 0 − (−4) = 4.
Solve 2x + y = 3, x + y = 2 by elimination of your choice, then check with the column picture.
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Subtract: x = 1, then y = 1. Columns: 1·(2, 1) + 1·(1, 1) = (3, 2) ✓