The mechanical rule is to dot each row of the matrix with the vector. That gives one entry of the answer per row.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The better view
Ax is a linear combination of the columns of A, with the entries of x as the weights. This is the view worth carrying forward.
Why it matters
Read that way, Ax = b asks whether b lies in the span of the columns of A. A whole chapter of ideas collapses into one question.
Sizes must match
An m by n matrix needs a vector with n entries and returns one with m. The inner numbers must agree.
Where the basis vectors go
Multiplying by ⟨1, 0⟩ picks out the first column. The columns are literally where the basis vectors land.
It is linear
A(u + v) = Au + Av and A(cu) = c(Au). Those two rules are what the word linear means.
A weighted recipe of the columns
Ax is a linear combination of the columns of A, with the entries of x as the weights. That reading is far more illuminating than the row-by-row dot product description.
Solvability is a span question
Ax = b has a solution exactly when b lies in the span of A's columns. The column view turns a computational question into a geometric one.
Dimensions must match
An m by n matrix takes vectors with n entries and produces vectors with m entries. Checking the shapes before computing catches most setup errors immediately.
It is a linear map
A(u + v) = Au + Av and A(cu) = cAu. Those two properties define linearity, and they are why matrices can represent every transformation the subject studies.
Step 2: Try It Yourself
Tap and try it out.
- Determinant of A5
- A times the vector(2, 3)
A non-zero determinant means the matrix is invertible. Its size is the factor by which areas are scaled — here 5.
Step 3: Watch an Example
One step at a time.
Watch Nour Multiply Two Ways
Nour computes A times ⟨2, 3⟩ where A has columns ⟨1, 4⟩ and ⟨5, 0⟩.
- Step 1
She uses the column view, weighting the first column by 2.
Step 4: Your Turn
Practice makes it stick.
The First Column
Problem 1 of 2
A has columns ⟨3, 7⟩ and ⟨1, 2⟩. What is the first entry of A times ⟨1, 0⟩?
The Size
Problem 2 of 2
A 3 by 4 matrix times a vector. How many entries must that vector have?
Weight the Columns
1 of 8
A has columns ⟨2, 0⟩ and ⟨0, 5⟩. What is the first entry of A times ⟨3, 1⟩?
2 of 8
The same A. What is the second entry of A times ⟨3, 1⟩?
3 of 8
A times ⟨0, 1⟩ gives which column of A? Enter 1 or 2.
4 of 8
A times the zero vector. What is every entry of the result?
5 of 8
A 2 by 5 matrix times a suitable vector. How many entries does the answer have?
6 of 8
Row 1 of A is [4, 2] and x = ⟨3, 1⟩. What is the first entry of Ax?
7 of 8
Match each product to its result.
Tap a card on the left to start.
8 of 8
A has columns ⟨1, 1⟩ and ⟨1, 1⟩. What is the second entry of A times ⟨2, 3⟩?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A has columns ⟨2, 1⟩ and ⟨0, 3⟩. What is the first entry of A times ⟨4, 5⟩?
Question 2 of 2
What is Ax, viewed the useful way?
What You Learned
- Ax can be computed row by row, but it is best understood as a combination of the columns of A.
- The entries of x are the weights, so Ax = b asks whether b lies in the span of the columns.
- Multiplying by a basis vector returns the matching column of A.