Section 10 Week 2 Learning Goals

Here are the knowledge and skills you should master by the end the second week.

10.1 Solution Sets, Span and Linear Independence

I should be able to do the following tasks:

Go back and forth between (i) systems of equations, (ii) vector equations, and (iii) the matrix equation \(Ax = b\).
Compute and understand the matrix-vector product \(A x\) both as a linear combination of the columns of A and as the dot product of \(x\) with the rows of \(A\).
Write the solution set to \(Ax=b\) as a parametric vector equation.
Determine whether a set of vectors is linearly dependent or independent
Find a dependence relation among a set of vectors
Decide if a set of vectors span \(\mathbb{R}^n\)

I should know and be able to use and explain the following terms or properties.

I should understand and be able to explain the following concepts:

Theorem 4 in Section 1.4 which says that the following are equivalent (they are all true or are all false) for an \(m \times n\) matrix \(A\)
- For each \(b \in \mathbb{R}^m\), the system \(A x = b\) has at least one solution
- Each \(b \in \mathbb{R}^m\) is a linear combination of the columns of \(A\)
- The columns of \(A\) span \(\mathbb{R}^m\)
- \(A\) has a pivot in every row.
Understand the relation between homogeneous solutions and nonhomogeneous solutions.
Linear independence
Span
More than \(n\) vectors in \(\mathbb{R}^n\) must be linearly dependent.