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Cogito

Linear Algebra · Chapter 1 · Lesson 2

Practice Worksheet

Linear Combinations and Span

Everything you can reach from where you started.

12 problems · about 23 minutes · N-VM.B.4, N-VM.B.5

NameDate

Figure — use these to answer the problems

The two arrows are your building blocks. Make them parallel and the shaded region collapses to a line: their span has lost a dimension.
ij
  • i-hat lands on(2, 1)
  • j-hat lands on(1, 3)
  • Determinant5
u, first entry2
v, first entry1
u, second entry1
v, second entry3
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. ⟨4, 8⟩ and ⟨1, 2⟩. What is the dimension of their span?

    Answer

  2. What is the span of a set of vectors?

    • Every point reachable by scaling and adding those vectors.
    • The longest of the vectors.

  3. 3⟨2, 1⟩ + 2⟨0, 5⟩. What is the second entry?

    Answer

  4. The span of a single non-zero vector. What is its dimension?

    Answer

  5. ⟨1, 0⟩ and ⟨0, 1⟩. What is the dimension of their span?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. ⟨2, 4⟩ and ⟨1, 2⟩. What is the dimension of their span?

    Answer

  2. Does the origin belong to every span? 1 yes, 0 no.

    Answer

  3. The span of the zero vector alone. What is its dimension?

    Answer

Stretch Yourself

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

4 problems

  1. Sort each pair of vectors by what they span.

    Write each item under the heading it belongs to: ⟨1, 0⟩ and ⟨0, 1⟩ · ⟨1, 1⟩ and ⟨2, 2⟩ · ⟨3, 1⟩ and ⟨1, 4⟩ · ⟨2, 6⟩ and ⟨1, 3⟩

    The whole plane

    Only a line

  2. 0⟨5, 7⟩ + 0⟨2, 9⟩. What is the first entry?

    Answer

  3. The Combination

    2⟨1, 3⟩ + 4⟨2, 0⟩. What is the first entry?

    Answer

  4. The Line

    Two vectors that are multiples of each other. What is the dimension of their span?

    Answer