Multiplying a matrix by a vector
A matrix A is a table of numbers, in rows and columns. To multiply it by a vector x = x₁x₂, there are two ways, and they always give the same answer. By rows: dot each row of A with x (slide 2), and each dot product gives one number of the answer. By columns: take x₁ of column 1, plus x₂ of column 2, a linear combination like on slide 1. Rows are the quick way to compute. Columns are the way to understand what Ax can be.
Lesson 1 · slide 3Matrix times vector- Row way: entry i of Ax is row i of A dotted with x.
- Column way: Ax = x₁·(column 1) + x₂·(column 2), a combination of the columns.
- Both ways add up the same products, so they always agree.
- x needs one number per column of A. Ax has one number per row of A.
A note on writing: aij is the entry in row i, column j, so a₁₂ is in row 1, column 2. That is what the small numbers in the banner above mean.
One example, two ways
Call the matrix A, and the two numbers in x x₁ and x₂. Take
Way 1, by rows. Each row of A is dotted with x. Row 1 gives the top number of the answer, row 2 the bottom number:
Way 2, by columns. x = 31 says: take 3 of column 1 and 1 of column 2, and add:
Same answer, 54. It has to be: both ways add the same four products, just in a different order. On the plane, the column way is a walk: 3 steps of 21, then 1 step of −11.
An everyday picture: a shopping basket
An egg has about 6 g of protein and 1 g of carbohydrate. A bowl of rice has about 4 g and 45 g. Each food is a column:
The basket holds 2·61 + 1·445 = 1647: 16 g of protein, 47 g of carbohydrate. That total is Ax. Adding whole foods, one column at a time, is the column way.
The basket by rows
Or add one nutrient at a time. Protein: 6·2 + 4·1 = 16. Carbohydrate: 1·2 + 45·1 = 47. Each is a dot product of a row of A with x.
Both ways add the same products, so they always agree. Rows are quicker by hand. Columns show what Ax can be.
Watch the sizes
A vector with n entries is an n by 1 matrix. x needs one entry per column of A, and Ax has one entry per row. So a 2 by 3 matrix (2 rows, 3 columns) takes in 3 numbers and gives back 2.
A bigger example: 4 nutrients, 3 foods
Add an apple, and track fat and fiber too. Now A has 4 rows (nutrients) and 3 columns (foods):
16 g protein, 72 g carbs, 10 g fat, 5 g fiber. Same idea, bigger table: 3 numbers go in, 4 come out, because (4 × 3) · (3 × 1) → (4 × 1). A matrix can have any number of rows and columns.
Next: So one x gives one output Ax. Next: try every x and collect all the outputs. That set tells us which targets b (vectors we want Ax to equal) we can hit.
Examples
Example 1 · Both ways
A = 3125, x = 12.
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Example 2 · A wide matrix
A = 102311 has 3 columns, so x needs 3 entries. Take x = 211. (2 × 3) · (3 × 1) → (2 × 1)
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3 numbers in, 2 numbers out: one per row of A.
Example 3 · Picking out a column
For any 2 × 2 matrix A: what is A times x = 10? And times 01?
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So the columns of A are exactly what A does to 10 and 01. We use this again in lesson 6, where a matrix moves the whole plane.
Example 4
Which x makes Ax equal to column 2 of A? Which x gives column 1 + column 2?
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x = 01 picks out column 2. x = 11 gives column 1 + column 2.
Example 5
A is 3 by 2. How many entries must x have, and how many does Ax have?
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x needs 2 entries, one per column. Ax has 3 entries, one per row.
Questions and comments
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