Let v, w be any vectors in ℝn. How does the span of {v, w} compare to the span of to the span of {v, w, 2v + 3w}?
Consider the matrix this 3x3 matrix where the last row is the sum of the first two. What's the echelon form?
In these 2 examples, there were some redudant information.
Definition: Let S = {u1, u2, …, um} be a set of vectors in ℝn. We say that S is linearly independent if the only if the only solution to the vector equation
x1u1 + x2u2 + … + xmum = 0
is the trivial solution - x1 = x2 = … = xm = 0. If a set if not linearly indepedent then it is linearly dependent.
A set is linearly dependent iff some vector is in the span of the others. A set is linearly independent iff no vector is in the span of the others.
Any set containing the zero vector is linearly dependent.
Example: Is the set {(16, 2, 8), (22, 4, 4), (18, 0, 4), (18, 2, 6)} linearly independent?
work out example in class using a linear system
Let S = {u1, …, um} be a set of vectors in ℝn and A = [u1 u2 … um] be the matrix formed by these vectors. Then S is linearly independent if and only if the only solution is the trivial solution.
Theorem: Let S = {u1, …, um} be a set of vectors in ℝn. Suppose
A = [u1 u2 … um]∼B,
where B is in echelon form. Then * S spans ℝn exactly when B has a pivot position in every row * S is linearly independent exactly when B has a pivot position in every column.
A set with fewer than n vectors will never span ℝn. A set with more than n vectors will never be linearly independent.
Let A be a matrix. Then A(x + y)=Ax + Ax and A(x − y)=Ax − Ay.
Example: Find a general solution for the linear system **** Using row reduction, we see that a general solution is of the form x = (1, 0, −5, 0)+s1(3, 1, 0, 0)+s2(−2, 0, 4, 1).
The solution to the homogenous system is x = s1(3, 1, 0, 0)+s2(−2, 0, 4, 1).
Let xp be a particular solution Ax = b. Then solutions have the form xg = xp + xh, where xp is a particular solution and xh is the general solution to the homogenous equations.
Theorem: Let A = [ai] and b be a vector in ℝn. Then the following are equivalent (if one is true then they are all true, if one is false then they are all false). * The set {a1, …, am} are linearly independent. * The vector equation x1a1 + x2a2 + … + xmam = b has at most one solution. * The linear system [a1 a2 … am|b] has at most one solution. * The equation Ax = b has at most 1 solution.
Example: Consider the vectors a1 = (1, 7, −2), a2 = (3, 0, 1), and a3 = (5, 2, 6). Set A = [ai]. Show that the columns of A are linearly independent and that Ax = b has a unique solution for every b in ℝ3.