Final Flashcards
Onto
- Span
* pivot in each row
1-to-1
- linearly indépendant
* pivot in each column
b is in the span {v1,v2,v3} iff
x1v1+x2v2+x3v3=b has a solution
{v1,v2,v3} are linearly independent iff
c1v1+c2v2+c3v3=0 has only the trivial solution (no free variables)
{0,v2,v3} is
linearly dependent
linear iff
Ax=0
plug in 0 for all x, if that gets you 0 then its linear
A set H in Rn is a subspace if it satisfies the following properties
1) the zero vector is in H
2) for any u v in H, u+v is in H
3) for any u in H cu is in H
A basis for a subspace H of Rn is:
A basis for a subspace H is a collection of vectors which are linearly independent and span H
Col(A)
The column space of A is the set of all linear combinations of the columns of A
Nul(A)
The null space of A is the set of all solutions to the homogenous equation Ax=0
The dimension of a subspace H is
the number of vectors in any basis of H
The rank of matrix A
is the dimension of the column space of A
what is the dimensions theorem
given an mXn matrix A the dimensions theorem states that dim(NulA) + rankA = n
how is the length (or norm) of a vector v defined by using inner product?
sqrt(v*v) the square root of the inner product of v with itself
A collection of vectors {u1,u2,…un} in Rn is said to be an orthogonal set if
ui*uj = 0 for any i not equal to j
