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Null space and column space
We will define matrix-vector multiplication and think about the set of vectors that satisfy Ax=0 for a given matrix A (this is the null space of A). We then proceed to think about the linear combinations of the columns of a matrix (column space). Both of these ideas help us think the possible solutions to the Matrix-vector equation Ax=b.
Defining and understanding what it means to take the product of a matrix and a vector
Showing that the Null Space of a Matrix is a valid Subspace
Calculating the null space of a matrix
Understanding how the null space of a matrix relates to the linear independence of its column vectors
Introduction to the column space of a matrix
Figuring out the null space and a basis of a column space for a matrix
Determining the planar equation for a column space in R3
Proof: Any subspace basis has same number of elements
Dimension of the Null Space or Nullity
Dimension of the Column Space or Rank
Showing that linear independence of pivot columns implies linear independence of the corresponding columns in the original equation
Showing that just the columns of A associated with the pivot columns of rref(A) do indeed span C(A).