I.1Multiplication Ax Using Columns of A

The one idea

You were probably taught to compute Ax a row at a time: take row 1, dot it with x, that’s the first entry of the answer. That works, and it is the wrong mental model for this book.

The view you want is by column. If A has columns a₁, a₂, …, aₙ and x = (x₁, x₂, …, xₙ), then

Ax = x₁a₁ + x₂a₂ + ⋯ + xₙaₙ

x is not a thing being consumed by rows. It is a list of weights, and Ax is the combination of columns those weights produce. The row picture gives you a number at a time; the column picture gives you the whole answer vector at once, as a recipe.

Why it matters later

Once Ax is “a combination of columns,” several later ideas stop being definitions you memorise and become obvious:

• The column space C(A) is just the set of every Ax you could possibly make. It is all the combinations of the columns. That is the definition, read straight off the formula above.
• Ax = b is solvable exactly when b lies in C(A). Not a separate theorem, a restatement. You can hit b if and only if b is reachable by mixing columns.
• Rank is the number of genuinely independent columns, meaning the number you actually need. Duplicated or dependent columns add nothing new to reach.
• A = CR, which Strang introduces here, falls out: put the independent columns in C, and R records how to rebuild every original column from them. The rank is the column count of C.

That last factorization is a warm-up for the pattern that runs through the whole book: take a matrix, split it into a small part carrying the real information and a part describing how to reassemble it. The SVD in I.8 is the same move done optimally.

What to actually do

1. Write a 3×3 matrix whose third column is the sum of the first two. Compute Ax for a couple of x vectors by combining columns, not by dotting rows. Notice you never escape a plane in 3-space.
2. For that matrix, write down C and R by hand. C has two columns; R is 2×3. Multiply them back and check you recover A.
3. State, without looking anything up, what C(A) is for your matrix: a line, a plane, or all of R³. And say why.

Check yourself

• If A is 5×3, what size is Ax, and where does that answer come from in the column picture?
• Why must the rank be at most the smaller of the two dimensions?
• A matrix has 7 columns and rank 4. How many columns does C have in A = CR? What does each row of R mean?

Common sticking points

“Isn’t this just the same arithmetic?” Yes, identical numbers, every time. The point is not a faster computation; it is that the column view makes the structure visible. Rank, column space, and solvability are all statements about columns, and they look like consequences rather than definitions once you read Ax this way.

Confusing C(A) with the space x lives in. For an m×n matrix, x has n entries and Ax has m. The column space sits in R^m: it is made of outputs. The nullspace, which arrives in I.3, sits in R^n with the inputs. Keeping those two straight early saves real confusion in Part I.

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This section is one of the three that MIT publishes in full, so you can read Strang’s own treatment rather than a summary of it. Do that, then come back and check your A = CR by hand.