Math 225 final review uiuc
Span( V1,...,VP)
set of all linear combinations
Basis of a vector subspace
Let H be a subspace of a vector space V. An indexed set B=(v1,...vp) in V is a basis for H if:
(i) B is a linearly independent set
(ii) the subspace spanned by B coincides with H,
that is, H=span(v1,...,vp)
Linearly Independent
c1V1+c2V2+...cpVp=0 has only trivial solution (c1,....,cp=0)
Linearly dependent
c1V1+c2V2+...cpVp=0 such that c1,...cp not all to 0
Definition of a Transpose of a matrix
An nxm matrix A^T whose columns are the corresponding rows of the mxn matrix A.
(A^T)^T=
A
, (A+B)^T
A^T+B^T
(rA)^T
rA^T
(AB)^T
B^TA^T
A^(-1)
1/detA (adjA)
det(A)
ad-bc (cross product)
adjA
(cij)^T
solve Ax=b (using A^(-1))
x=A^(-1)b
(A^-1)^-1
Span( V1,...,VP)
set of all linear combinations
Basis of a vector subspace
Let H be a subspace of a vector space V. An indexed set B=(v1,...vp) in V is a basis for H if:
(i) B is a linearly independent set
(ii) the subspace spanned by B coincides with H,
that is, H=span(v1,...,vp)
Linearly Independent
c1V1+c2V2+...cpVp=0 has only trivial solution (c1,....,cp=0)
Linearly dependent
c1V1+c2V2+...cpVp=0 such that c1,...cp not all to 0
Definition of a Transpose of a matrix
An nxm matrix A^T whose columns are the corresponding rows of the mxn matrix A.
(A^T)^T=
A
, (A+B)^T
A^T+B^T
(rA)^T
rA^T
(AB)^T
B^TA^T
A^(-1)
1/detA (adjA)
det(A)
ad-bc (cross product)
adjA
(cij)^T
solve Ax=b (using A^(-1))
x=A^(-1)b
(A^-1)^-1
