What you can now do.
Four ideas from Strang chapter 1, all built on one operation: stretch and add.
Lesson 1 · slide 5Recap- Linear combinations cv + dw are the basic operation.
- The dot product v · w multiplies matching parts and adds. It gives lengths, and v · w = 0 means a right angle.
- Ax by rows: each row dotted with x. Ax by columns: a combination of the columns. Same answer.
- The column space C(A) holds every Ax, and Ax = b is solvable exactly when b is in it.
- §1.1 Linear combinations. Stretch, then add: steps of v plus steps of w.
- §1.2 Length and dot product. Multiply matching parts and add. ‖v‖ = √(v · v). Zero means a right angle.
- §1.3 Matrix times vector. Compute by rows (dot products), think by columns (a combination). The basket: each column is a food, x says how many of each, Ax is the total.
- §1.3 Column space. Every spot your steps can reach. Ax = b is solvable exactly when b is in C(A).
Next lesson: solving Ax = b (Strang chapter 2)
First, two ways to read a system of equations: as lines that cross, or as columns that combine to b. Then solve it the way a computer does: subtract multiples of one row from another until the matrix is a triangle, and solve from the bottom up. Along the way: how to multiply two matrices, and a way to save the elimination steps so the next system with the same matrix is quick.
Read along
Professor Gilbert Strang, Introduction to Linear Algebra, 5th edition (2016), chapter 1. His MIT 18.06 lectures are free on MIT OpenCourseWare.
Examples
Example 1
A = 1326. Describe C(A). Is 24 in it?
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Column 2 = 3·column 1, so C(A) is the line through 12. 24 = 2·12 is on it, for example x = 20.
Example 2
Find v · w and the lengths for v = 122 and w = 20−1. Are they perpendicular?
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v · w = 2 + 0 − 2 = 0, so yes. ‖v‖ = √(1 + 4 + 4) = 3 and ‖w‖ = √(4 + 0 + 1) = √5.
Example 3
Compute Ax both ways for A = 2111 and x = 11.
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Rows: 2·1 + 1·1 = 3 and 1·1 + 1·1 = 2. Columns: 1·21 + 1·11 = 32. Same answer, 32.
Questions and comments
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