Dimension and rank

Intros
Lessons
1. Dimension and Rank Overview:
2. Dimension of a subspace
• Dimension = number of vectors in the basis
• Can we find dimension of column space and null space?
3. Rank of a Matrix
• Find the basis
• Count the # of vectors
• Shortcut = count the # of pivots
4. Dimension of the Null Space
• Find the general solution
• Put in parametric vector form
• Count the # of vectors
• Shortcut = count the # of free variables
5. The Rank Theorem
• Rank $A +$ dim $N(A) = n$
• An example of using the theorem
Examples
Lessons
1. Finding the Rank of a matrix
Find the rank of $A$ if:
1. Find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace?
1. Finding the dimension of the null space
Find the dimension of the null space of $A$ if:
1. Utilizing the Rank Theorem
You are given the matrix $A$ and the echelon form of $A$. Find the basis for the column space, and find the rank and the dimensions of the null space.
1. Understanding the Theorems
Let the subspace of all solutions of $Ax=0$ have a basis consisting of four vectors, where $A$ is $4 \times 6$. What is the rank of $A$?
1. Let $A$ be a $m \times n$ matrix where the rank of $A$ is $p$. Then what is the dimension of the null space of $A$?