4- Vector Spaces Flashcards
What is a vector space?
Not including the 10 axioms
A nonempty set V of vectors on which are defined two operations, called addition and multiplication by scalars (real numbers)
What are the ten axioms of a vector space?
Where vectors u, v, and w are in V and c and d are real scalars
1) u + v is in V
2) u + v = v + u
3) (u + v) + w = u + (v + w)
4) 0 exists in V, such that u + 0 = u
5) u + (-u) = 0
6) cu exists in V
7) c(u + v) = cu + cv
8) (c + d)u = cu + du
9) c(du) = (cd)u
10) 1*u = u
What is a subspace?
Including the three axioms
Subspace (H) of a vector space V has these properties 1) the zero vector of V is in H 2) H is closed under addition For each u and v in H, u + v is in H 3) H is closed under multiplication. For each scalar c, cu is in H
What is the least amount of subspaces that a vector space has? What are they?
2
Zero subspace, or the trivial subspace
Itself
What is theorem 1 regarding subspaces?
If v1,…,vp are in a vector space V, then Span{v1,…,vp} is a subspace of V
What is the null space?
The null space of an mxn matrix A (Nul A) is the set of all solutions of the homogeneous equation Ax = 0
What is theorem 2? Regarding null space and subspace
The null space of an mxn matrix A is a subspace of R^n. Equivalently, the set of all solutions to Ax = 0 is a subspace of R^n
How do we know if a set spans R^n?
Pivot in every row
What is the definition of a column space?
Col A is the set of all linear combos of the columns of A, then
Col A = Span {a1, … , an}
True or false. Why?
The column space of an mxn matrix is a subspace of R^m
True.
This is literally theorem 3
4 biggest contrasts between Nul A and Col A
Nul A
1) Nul A is a subspace of R^n
2) x is in Nul A iff Ax = 0
3) Nul A = {0} iff Ax = 0 has only the trivial solution
4) Nul A = {0} iff linear transformation x |-> Ax is one-to-one
Col A
1) Col A is a subspace of R^m
2) b is in Col A iff Ax = b
3) Col A = R^m iff Ax = b has a solution for all b that’s in R^m
4) Col A = R^m iff the linear transformation x |-> Ax maps
R^n onto R^m
What is the kernel and range of a linear transformation?
Kernel is Nul A
Range is Col A
What is the definition of a basis?
Let H be a subspace of a vector space V. B = {b1, … ,bp} in V is a basis of H if
(i) B is a linearly independent set, &
(ii) H = Span {b1, … , bp}
What is the spanning set theorem?
Let S = {v1, … , vp} be a set in V and let H = Span {v1, … , vp}
(a) take out any vectors in S that are linearly dependent of the other vectors until your entire set is linearly independent
(b) if H =/= {0}, some subset of S is a basis for H
True or false. Why?
The pivot columns of a matrix A form a basis for Col A
True
Literally theorem 6
