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Lesson 2: Subspaces, Span, and Linear Combinations

Estimated time: 35-45 minutes

Learning Objectives

Linear Combinations

Linear Combination: A vector v is a linear combination of v_1, v_2, ..., v_k if v = c_1 v_1 + c_2 v_2 + ... + c_k v_k for some scalars c_1, ..., c_k.

Example

Is (7, 4, -1) a linear combination of (1, 0, 1) and (2, 1, -1)?

We need c_1(1,0,1) + c_2(2,1,-1) = (7,4,-1). This gives the system: c_1 + 2c_2 = 7, c_2 = 4, c_1 - c_2 = -1.

From equation 2: c_2 = 4. From equation 3: c_1 = -1 + 4 = 3. Check equation 1: 3 + 8 = 11 is not 7.

No, (7,4,-1) is NOT a linear combination of those vectors (the system is inconsistent).

Span

Span: The span of vectors v_1, ..., v_k is the set of ALL possible linear combinations: Span{v_1,...,v_k} = {c_1 v_1 + ... + c_k v_k : c_1,...,c_k in R}.

Geometrically: span of one nonzero vector = a line through the origin. Span of two non-parallel vectors = a plane through the origin. Span of three independent vectors in R^3 = all of R^3.

Example

Span{(1,0), (0,1)} = all of R^2 (every vector (a,b) = a(1,0) + b(0,1)).

Span{(1,2), (2,4)} = a line through the origin (since (2,4) = 2(1,2), they are not independent).

Subspaces

Subspace: A subset H of R^n is a subspace if: (1) The zero vector 0 is in H. (2) H is closed under addition: if u, v are in H, then u + v is in H. (3) H is closed under scalar multiplication: if v is in H and c is a scalar, then cv is in H.

Examples of Subspaces

  • {0} (just the zero vector) -- always a subspace
  • R^n itself -- always a subspace
  • Span{v_1, ..., v_k} for any set of vectors -- always a subspace
  • The null space of a matrix (solutions of Ax = 0)
  • The column space of a matrix (all vectors b such that Ax = b is consistent)

Non-Example

The set {(x, y) : x ≥ 0} (right half-plane) is NOT a subspace. It contains (1, 0) but not (-1)(1, 0) = (-1, 0). It fails closure under scalar multiplication.

Column Space and Null Space

Column Space Col(A): The span of the columns of A. Equivalently, {b : Ax = b has a solution}.

Null Space Nul(A): The set of all solutions to Ax = 0. That is, Nul(A) = {x : Ax = 0}.

Both are subspaces. The null space lives in R^n (where n = number of columns). The column space lives in R^m (where m = number of rows).

Check Your Understanding

1. Express (5, 1) as a linear combination of (1, 0) and (3, 1).

Answer: c_1(1,0) + c_2(3,1) = (5,1). From component 2: c_2 = 1. From component 1: c_1 + 3 = 5, c_1 = 2. So (5,1) = 2(1,0) + 1(3,1).

2. Is the set {(x,y) : y = 2x} a subspace of R^2?

Answer: Yes. (1) Zero: (0,0) satisfies 0=2(0). (2) Closed under +: if y_1=2x_1 and y_2=2x_2, then y_1+y_2=2(x_1+x_2). (3) Closed under scalar mult: c(y)=2(cx). This is the line y=2x through the origin.

3. Is {(x,y) : x + y = 1} a subspace of R^2?

Answer: No. (0,0) does not satisfy 0+0=1. The zero vector is not in the set, so it cannot be a subspace.

4. What is the null space of I_n (the identity matrix)?

Answer: Nul(I_n) = {0}. The only solution to Ix = 0 is x = 0.

Key Takeaways

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