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Lesson 3: Linear Independence

Estimated time: 35-45 minutes

Learning Objectives

Define linear independence and dependence
Test for linear independence by solving a homogeneous system
Relate independence to pivots, determinants, and invertibility
Apply key facts about sets with more vectors than dimensions

Definition

Linearly Independent: Vectors v_1, ..., v_k are linearly independent if the only solution to c_1 v_1 + c_2 v_2 + ... + c_k v_k = 0 is c_1 = c_2 = ... = c_k = 0 (the trivial solution).

Linearly Dependent: The vectors are dependent if there exist scalars, not all zero, such that c_1 v_1 + ... + c_k v_k = 0. Equivalently, one vector can be expressed as a linear combination of the others.

How to Test

Form a matrix A whose columns are the vectors. Row reduce. The vectors are linearly independent if and only if every column is a pivot column (i.e., the only solution to Ax = 0 is x = 0).

Worked Example: Independent

Test {(1,0,0), (0,1,0), (0,0,1)}. Matrix: I_3. RREF = I_3. Three pivot columns. Linearly independent.

Worked Example: Dependent

Test {(1,2), (3,6)}. Matrix: [1 3; 2 6]. R_2 - 2R_1: [0 0]. RREF: [1 3; 0 0]. Only 1 pivot, but 2 vectors. Free variable exists. Linearly dependent.

Dependence relation: (3,6) = 3(1,2), or equivalently 3(1,2) - 1(3,6) = (0,0).

Key Facts

Important Results
        A set containing the zero vector is always linearly dependent.
A set of more than n vectors in R^n is always linearly dependent.
For square matrices: columns are independent iff det is not zero iff the matrix is invertible.
A set of two vectors is dependent iff one is a scalar multiple of the other.

      

Check Your Understanding

1. Are (1,1,0), (0,1,1), (1,0,-1) linearly independent?

Answer: Form A = [1 0 1; 1 1 0; 0 1 -1]. det(A) = 1(1*(-1)-0*1) - 0 + 1(1*1-1*0) = -1+1 = 0. det = 0, so they are linearly dependent. In fact, (1,0,-1) = (1,1,0) - (0,1,1).

2. Can 4 vectors in R^3 be linearly independent?

Answer: No. More vectors than the dimension guarantees dependence.

3. If {v_1, v_2, v_3} is linearly independent and v_4 is NOT in Span{v_1, v_2, v_3}, is {v_1, v_2, v_3, v_4} independent?

Answer: Yes (assuming we are in R^n with n ≥ 4). If v_4 is outside the span, it cannot be written as a combination of the others, preserving independence.

Key Takeaways

Independent: only the trivial solution makes c_1 v_1 + ... + c_k v_k = 0
Test: form matrix with vectors as columns, row reduce, check for free variables
More than n vectors in R^n are always dependent
Columns independent iff every column is a pivot column iff det is nonzero (square case)

Lesson 4: Basis and Dimension

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