Reading Book Linear Algebra (Pg.138, 140 of 584)
This text is explaining one of the most fundamental concepts in linear algebra: The Span of a set of vectors.
Here is a breakdown of what it’s saying in plain English:
1. The Starting Point: A Set of Vectors (\(S\))¶
Imagine you have a handful of vectors (arrows) in space. This collection is called \(S\). * Usually, \(S\) is just a "list" of vectors. It doesn't have to be a full, infinite subspace. It’s just your raw material.
2. The Process: Linear Combinations¶
The text says that to get to \(SS\), you take "all combinations" of those vectors. This means you take every vector in your list, multiply them by any numbers you want (constants \(c_1, c_2, \dots\)), and add them together. * Recipe: \(c_1v_1 + c_2v_2 + \dots + c_Nv_N\) * Because you can choose any numbers for \(c\), you end up creating an infinite number of new vectors.
3. The Result: A Subspace (\(SS\) or the "Span")¶
When you collect every possible combination you can make, you have "filled in the gaps" between your original vectors. This new, larger collection is called the Span of \(S\). * It is "definitely a subspace": This is because if you add two combinations together, or scale one, you just get another combination. It stays within its own "territory." * It is the "smallest" subspace: This means that if you want to build a subspace that contains all the vectors in your original set \(S\), the Span is the absolute minimum amount of "territory" you need to cover. You can't make it any smaller without breaking the rules of a subspace.
Real-World Examples mentioned in the text:¶
- A single vector: If your set \(S\) is just one arrow pointing out from the origin, its "span" is the entire infinite line that the arrow sits on.
- Columns of a matrix: If you take the columns of a matrix and find their span, you have created the Column Space. This is why we study the columns of a matrix—to see what kind of "territory" (subspace) they can reach.
Summary¶
The text wants you to understand that you can "manufacture" a subspace just by picking a few vectors and looking at every possible way to combine them. That "territory" you create is called the Span.
This text is explaining one of the most fundamental concepts in linear algebra: The Span of a set of vectors.
Here is a breakdown of what it’s saying in plain English:
1. The Starting Point: A Set of Vectors (\(S\))¶
Imagine you have a handful of vectors (arrows) in space. This collection is called \(S\). * Usually, \(S\) is just a "list" of vectors. It doesn't have to be a full, infinite subspace. It’s just your raw material.
2. The Process: Linear Combinations¶
The text says that to get to \(SS\), you take "all combinations" of those vectors. This means you take every vector in your list, multiply them by any numbers you want (constants \(c_1, c_2, \dots\)), and add them together. * Recipe: \(c_1v_1 + c_2v_2 + \dots + c_Nv_N\) * Because you can choose any numbers for \(c\), you end up creating an infinite number of new vectors.
3. The Result: A Subspace (\(SS\) or the "Span")¶
When you collect every possible combination you can make, you have "filled in the gaps" between your original vectors. This new, larger collection is called the Span of \(S\). * It is "definitely a subspace": This is because if you add two combinations together, or scale one, you just get another combination. It stays within its own "territory." * It is the "smallest" subspace: This means that if you want to build a subspace that contains all the vectors in your original set \(S\), the Span is the absolute minimum amount of "territory" you need to cover. You can't make it any smaller without breaking the rules of a subspace.
Real-World Examples mentioned in the text:¶
- A single vector: If your set \(S\) is just one arrow pointing out from the origin, its "span" is the entire infinite line that the arrow sits on.
- Columns of a matrix: If you take the columns of a matrix and find their span, you have created the Column Space. This is why we study the columns of a matrix—to see what kind of "territory" (subspace) they can reach.
Summary¶
The text wants you to understand that you can "manufacture" a subspace just by picking a few vectors and looking at every possible way to combine them. That "territory" you create is called the Span.
The author is using \(V_2\) to illustrate how you can create a "nest" of subspaces—starting with a large space and narrowing it down into smaller and smaller pieces.
Here is exactly what the author is saying about \(V_2\):
1. The Big Space (\(V_2\))¶
The problem defines \(V_2\) as all vectors perpendicular to \(u = (1, 2, 1)\). * What it is: Geometrically, \(V_2\) is a plane passing through the origin in 3D space. Any vector on this plane will satisfy the equation \(1x + 2y + 1z = 0\). * Example: The vector \((1, -1, 1)\) is in \(V_2\) because \(1(1) + 2(-1) + 1(1) = 0\).
2. The Subspace (\(S\))¶
The author then picks a smaller subspace inside that plane. * How they do it: They choose just one specific vector that lives on that plane, like \((1, -1, 1)\), and take all of its multiples. * What it is: This creates a line (\(S\)). Since this line is made of vectors from the plane, the line "lives" entirely inside the plane. So, \(S\) is a subspace of \(V_2\).
3. The Tiny Subspace (\(SS\))¶
Finally, the author picks an even smaller subspace inside the line \(S\). * How they do it: They pick the zero vector \(x = (0, 0, 0)\). * What it is: The set of all multiples of the zero vector is just the zero vector itself. This is the Zero Subspace (labeled \(Z\) in the text). * The logic: Every vector space must contain the zero vector. Therefore, the point at the origin is the absolute smallest subspace possible. It lives on the line \(S\), which lives on the plane \(V_2\).
Summary of the "Chain"¶
The author is showing you a hierarchy of dimensions: 1. \(V_2\) (Dimension 2): A full plane (all solutions to \(x + 2y + z = 0\)). 2. \(S\) (Dimension 1): A single line sitting on that plane. 3. \(SS\) (Dimension 0): Just the origin point \((0, 0, 0)\) sitting on that line.
The phrase "So many possibilities" at the end of the first paragraph is the author's way of saying there is no "right" answer for \(S\) and \(SS\). You could have picked any other vector on the plane to make a different line \(S\), or any other vector on the line to make a different \(SS\). He just picked these to show how the "drilling down" process works.