References: Spans and basis

Source material

Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Linear Algebra, Chapter 2: "Linear combinations, span, and basis vectors"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/span
  Series index: https://www.3blue1brown.com/?topic=linear-algebra
  License: copyright Grant Sanderson; videos published on his site and YouTube
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Watch this next

• Linear combinations, span, and basis vectors (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors. Sanderson animates the span sweeping out as you turn the two scalars, so you watch the plane fill in and watch it collapse to a line the instant the vectors line up. About ten minutes. If the three span cases felt abstract in text, the animation makes them obvious.

Going deeper

• Essence of Linear Algebra (full series) by 3Blue1Brown. The series this track follows. The previous chapter (vectors) and the next (Linear transformations and matrices) sit on either side of this one.

• Khan Academy: Vectors and spaces for a slower, exercise-driven walk through the same ideas. The “Linear combinations and spans” unit covers exactly this lesson’s ground, with practice problems and immediate feedback.

Adjacent topics

Where this sits in the track.

• What a vector actually is (previous lesson). Span is built entirely on the two operations introduced there, adding and scaling. If “linear combination” felt slippery, it is just those two operations used together, so a quick reread of the previous lesson grounds it.

• Linear transformations and matrices (next lesson). Once you know what a basis is, the natural next question is what happens to the whole space when you move the basis vectors. That single idea, “decide where i-hat and j-hat land,” turns out to define every linear transformation, and a matrix is just the record of where they went.