Linear eqn form
What is a solution set of linear eqns
What is row equivalence
When one matrix can be turned into the other via row opertations
What happens if two augmented matrices are row equivalent?
Then they have the same soln set.
How many matrices can one matrix be row equivalent to in RREF
Only one matrix
Pivot position/Pivot column
What is a basic and a free variable
What makes a linear system consistent? What is a unique soln? What is infinitely many solutions?
What is a span?
What is an elementary matrix? What is a permuatation matrix?
What is the product of two lower (upper) triangular matrices?
The product is also lower (upper)
When does a matrix A have an LU-Decomp?
What are the L and U linear systems?
What is a permuation matrix?
Its an elementary matrix comprised of elementary row operations to allow a matrix have an LU- Decomp
What is dot product
v*w = v^T(w)
The norm of v is
What is pairwise orthogonal?
What makes a subset a subspace?
It must be closed under vector addition and scalar multiplication
What makes a span a subspace?
What is a column space?
For linear system Ax = b. Is b in the col(A)?
Yes, if the linear system Ax = b has a solution.
What is a nullspace?
The set of all solutions to the linear system Ax = 0
Is the null space a subspace?
Yes, it is a subspace of the # of columns in the matrix.
Can a subset be a subspace for a vector space?
Yes, as long as the subset is closed under vector addition and scalar multiplication
What is linearly independence?
What is a basis?
What is a dimension of V?
The number of Vectors in a basis of V.
What does it mean for a basis to be a minimal spanning set?
The elements of the basis span(V) but you cannot delete any of these elements.
What is the rank of a matrix?
The number of pivots it has.
dimNul(A) if A is M x N
dimNul(A) = N - r (number of free variables)
dimCol(A) if A is M x N
dimCol(A) = r (# of pivots)
What is the col(A^T) and col(B^T) if they are row equivalent?
col(A^T) = col(B^T)
What do the non-zero rows in an echelon form matrix A form?
What is an orthogonal compliment?
Is Nul(A) an orthogonal compliment of Col(A^T)
How to express a vector w in V(vector space)
Coordinate Vector Definition
What is a standard basis
Change of base matrix
Solving for the original vector (v) if given the basis (B) and the coordinate vector (vB)
Computing
Computing
What is a Orthogonal Basis
Solving for a vector using an orthogonal basis
What makes an N x N matrix orthogonal?
What is a linear transformation?
This shows that you can apply T to v and w first then apply a and b. where a and b are scalars
What values completely determine T if T is a linear transformation? (Where v1,...vn, is a basis of V) (Vector Spaces)
How can we break down the liner transformation T into a matrix A and the standard basis E?
How to change linear transformations into matrix multiplication
How to go from one base to another, change dimensions using a linear transformation then go from that base to another base
