Matrix Times Vector

A weighted recipe of the columns.

12 problems · about 22 minutes · N-VM.C.11, N-VM.C.12

What this lesson teaches

The student multiplies a matrix by a vector and interprets the result as a combination of columns.

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.

Straightforward practice. Get the method working first.

5 problems

A has columns ⟨2, 1⟩ and ⟨0, 3⟩. What is the first entry of A times ⟨4, 5⟩?
Answer 8
Why 8.
What is Ax, viewed the useful way?
Answer A linear combination of the columns of A, weighted by the entries of x.
Why A weighted recipe of the columns.
A has columns ⟨2, 0⟩ and ⟨0, 5⟩. What is the first entry of A times ⟨3, 1⟩?
Answer 6
Why 3 times 2, plus 1 times 0.
The same A. What is the second entry of A times ⟨3, 1⟩?
Answer 5
Why 3 times 0, plus 1 times 5.
A times ⟨0, 1⟩ gives which column of A? Enter 1 or 2.
Answer 2
Why The weight sits on the second column.

The same ideas with more to keep track of.

3 problems

A times the zero vector. What is every entry of the result?
Answer 0
Why All the weights are zero.
A 2 by 5 matrix times a suitable vector. How many entries does the answer have?
Answer 2
Why One per row.
Row 1 of A is [4, 2] and x = ⟨3, 1⟩. What is the first entry of Ax?
Answer 14
Why 12 + 2.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each product to its result.
Answer A times ⟨1, 0⟩ → The first column of A; A times ⟨0, 1⟩ → The second column of A; A times ⟨0, 0⟩ → The zero vector
Why The weights select columns.
A has columns ⟨1, 1⟩ and ⟨1, 1⟩. What is the second entry of A times ⟨2, 3⟩?
Answer 5
Why 2 + 3.
The First Column: A has columns ⟨3, 7⟩ and ⟨1, 2⟩. What is the first entry of A times ⟨1, 0⟩?
Answer 3
Why 3.
The Size: A 3 by 4 matrix times a vector. How many entries must that vector have?
Answer 4
Why 4.