Practice: Spans and basis
Self-check
Section titled “Self-check”Six short questions. Answer each one in your head (or on paper) before opening the collapsible. Trying to retrieve the answer is where the learning sticks; rereading feels productive but does much less.
1. What is a linear combination of two vectors v and w?
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Any expression a · v + b · w where a and b are scalars you choose freely. It is just the two operations from the previous lesson used together: scale each vector, then add the results.
2. What is the span of a set of vectors?
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The set of all linear combinations of them: every point you can reach by scaling each vector by any amount and adding the results. Not the vectors themselves, but everything reachable from them.
3. For two vectors in the plane, what are the three things the span can be?
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The whole plane (if the two point in different directions, that is, are independent), a single line through the origin (if one is a scalar multiple of the other), or just the origin (if both are the zero vector).
4. What two properties make a set of vectors a basis?
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It must be linearly independent (no redundancy; no vector sits in the span of the others) and it must span the whole space (no gaps; every point is reachable). Independent plus spanning: the smallest kit that builds everything.
5. What is the difference between a linearly dependent and a linearly independent set?
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A set is dependent if at least one vector is already in the span of the others, so it could be dropped without shrinking the span (it is redundant). It is independent when every vector adds a genuinely new direction the others cannot reach.
6. Why does every span pass through the origin?
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Because choosing all scalars equal to zero (a = 0, b = 0, …) always produces the zero vector. The origin is reachable from any set of vectors, which is why spans are lines and planes through the origin, never off to the side.
Try it yourself, part 1: classify the span
Section titled “Try it yourself, part 1: classify the span”For each pair of vectors, decide whether the span is the whole plane, a single line, or just the origin. The test is simple: is one vector a scalar multiple of the other (dependent, so a line), do they point in different directions (independent, so the plane), or are they both zero (the origin)? About 7 minutes, pen and paper.
- a)
[1, 0]and[0, 1] - b)
[2, 3]and[4, 6] - c)
[1, 2]and[2, 1] - d)
[3, -1]and[-6, 2] - e)
[0, 0]and[0, 0]
Check your work
- a) The whole plane. Different directions (independent); this is the standard basis.
- b) A line.
[4, 6] = 2 · [2, 3], so the second is a multiple of the first (dependent). - c) The whole plane. Neither is a multiple of the other (
1/2is not2/1), so they point in different directions. - d) A line.
[-6, 2] = -2 · [3, -1], so they lie on the same line through the origin (dependent). - e) Just the origin. Both are the zero vector; every combination is still
[0, 0].
The quick check for two 2D vectors: they are dependent (span a line) exactly when one is a scalar multiple of the other. Everything else nonzero spans the plane.
Try it yourself, part 2: coordinates in a new basis
Section titled “Try it yourself, part 2: coordinates in a new basis”A basis lets every vector be written as a unique combination of the basis vectors, and the scalars are the coordinates in that basis. This drill makes that concrete. About 8 minutes.
Step 1. In the standard basis, the point [7, 1] is trivially 7 · [1, 0] + 1 · [0, 1]. Now express the same point in the basis B = {[1, 1], [1, -1]}. Find scalars a and b with a · [1, 1] + b · [1, -1] = [7, 1].
Step 2. Now try to express [7, 1] in the pair {[1, 1], [2, 2]}. What goes wrong, and what does it tell you about that pair?
Check your work
Step 1. Component by component, a · [1, 1] + b · [1, -1] = [7, 1] becomes a + b = 7 and a - b = 1. Adding the two equations gives 2a = 8, so a = 4, and then b = 3. Check: 4 · [1, 1] + 3 · [1, -1] = [4, 4] + [3, -3] = [7, 1]. One basis, one unique answer. Same point, different basis, different coordinates ([4, 3] instead of [7, 1]).
Step 2. [2, 2] = 2 · [1, 1], so the pair is dependent: both lie on the line y = x. Every combination a · [1, 1] + b · [2, 2] lands somewhere on that line. Since [7, 1] is not on y = x (7 is not 1), no choice of scalars reaches it. That is what dependence costs you: a whole dimension of reach, and the pair is not a basis for the plane.
Flashcards
Section titled “Flashcards”Ten cards. Click any card to reveal the answer. Use the Print flashcards button to lay out the full set as one card per page, ready to print or save as a PDF for offline review.
Q. What is a linear combination of vectors v and w?
Any a · v + b · w for chosen scalars a and b: scale each vector, then add the results. It is the two operations from the previous lesson (adding and scaling) used together.
Q. What is the span of a set of vectors?
The set of all their linear combinations: every point reachable by scaling each vector by any amount and adding. It is the reachable set, not the original vectors.
Q. What are the three possible spans of two vectors in the plane?
The whole plane (if they point in different directions), a single line through the origin (if one is a scalar multiple of the other), or just the origin (if both are the zero vector).
Q. What is a basis?
A set of vectors that is both linearly independent (no redundancy) and spans the whole space (no gaps). The smallest kit of vectors that can build every point in the space.
Q. Linearly dependent vs linearly independent?
Dependent: at least one vector is already in the span of the others, so it is redundant and can be dropped without shrinking the span. Independent: every vector adds a new direction the others cannot reach.
Q. Why does every span pass through the origin?
Setting all scalars to zero gives the zero vector, so the origin is always reachable. That is why spans are lines and planes through the origin, never shifted off to the side.
Q. How does the standard basis relate to coordinates?
Writing [3, 4] means 3 · [1, 0] + 4 · [0, 1], that is, 3 of i-hat plus 4 of j-hat. The coordinates are the amounts of each basis vector; uniqueness is what lets coordinates be coordinates.
Q. When do two vectors in the plane span the whole plane?
Exactly when they are linearly independent, meaning neither is a scalar multiple of the other. The moment one lines up with the other, the span collapses to a single line.
Q. What is the dimension of a space, in terms of a basis?
The number of vectors in a basis for it: how many independent directions it takes to reach every point. Each independent vector you add lifts the span by one dimension.
Q. Does adding more vectors always increase the span?
No. Once a set already spans the whole space, every extra vector lies in that span and is therefore dependent. Beyond a basis, more vectors add reach you already had.