Term
|
Definition
| a square matrix whose determinant is zero |
|
|
Term
| If A(inverse) exists then Ax=b has the unique solution: |
|
Definition
|
|
Term
| A nxn matrix is invertible if and only if... |
|
Definition
|
|
Term
|
Definition
| maximum number of independent rows/columns |
|
|
Term
| If AB is invertible, then... |
|
Definition
| both A and B are invertible |
|
|
Term
|
Definition
| defined only for square matrices |
|
|
Term
|
Definition
| determinant of the square matrix formed by deleting one row and one column from some larger square matrix |
|
|
Term
|
Definition
| add all of these to get the determinant |
|
|
Term
|
Definition
MOST IMPORTANT: If a subset S of a vector space V fails to contain the zero vector, then it cannot form a subspace
Next: check if it is closed under addition and then scalar multiplication |
|
|
Term
|
Definition
put in row echelon form and find columns with leading 1's
*need at least n vectors to span R^n and need to be linearly independent |
|
|
Term
| The vectors are linearly dependent if... |
|
Definition
the equation has a solution (equal to zero) then at least one of the scalars (ai) is not zero
determinant = 0
a set of n+1 or more vectors in R^n, since dim = n
if there are any free variables |
|
|
Term
| To check linear dependence/independence |
|
Definition
| put set of vectors in row echelon form |
|
|
Term
| A set of only one vector is linearly independent if and only if that one vector is... |
|
Definition
|
|
Term
| a set containing two vectors is linearly independent so long as... |
|
Definition
| one vector is not a multiple of the other |
|
|
Term
| a set containing the zero vector is always |
|
Definition
|
|
Term
|
Definition
# of vectors in any basis for V
= # of free variables |
|
|
Term
|
Definition
| ALWAYS: linearly independent and span V |
|
|
Term
| dimension of the column space = |
|
Definition
|
|
Term
|
Definition
| Let A be an nun matrix. The set of column vectors of A corresponding to those column vectors containing leading ones in any row-echelon form of A is a basis for colspace (A). |
|
|
Term
|
Definition
| The set of nonzero row vectors in any row-echelon form of an mxn matrix A is a basis for rowspace (A) |
|
|
Term
|
Definition
| find reduced row echelon form -> then find free variable -> vectors multiplied to those free variables are the span of the nullspace -> if linearly independent then a basis as well! |
|
|
Term
|
Definition
|
|
Term
| when giving example of bases of R^n |
|
Definition
| give standard, then multiply any of them with a number and still a basis! |
|
|
