finals: sections 2.9 - 4.5 Flashcards
what is a coordinate vector?
the b-coordinate vector is the vector that holds the weights for the linear combination x1v1 + x2v2 + … + xpvp = b
what is the dimension of a subspace?
the dimension of a nonzero subspace H, denoted dim H, is the number of vectors in any basis for H
what is the rank of a matrix?
Rank A is the dimension of the column space (# of nonzero rows in echelon form)
why do we learn about vector spaces?
vector spaces are more general than R^n.
is R^n a vector space?
yes
what is the dimension for a vector space?
dim V is the number of vectors in any basis for V
what is nullity?
dimension of the nullspace
what is a vector space?
a nonempty set of V objects, called vectors, on which are defined by two operations, called addition and multiplication by scalars, subject to the ten axioms. The axioms must hold for all vectors u, v, and w in V and for all scalars c and d
what are the ten axioms for vector spaces?
1) the sum of u and v, denoted u + v, is in V
2) u + v = v + u
3) (u + v) + w = u + (v + w)
4) for every vector u in V, there is a zero vector in V such that u + 0 = u
5) for every vector u in V, there is a vector -u in V such that u + (-u) = 0
6) the scalar multiple of u by c, denoted cu, is in V
7) c(u + v) = cu + cv
8) (c + d)u = cu + du
9) (cd)u = c(du)
10) 1u = u
A subspace of a vector space V is a subset H of V that has three properties:
1) the zero vector of V is in H
2) H is closed under vector addition. for each u and v in H, the sum u + v is in H
3) H is closed under scalar multiplication. for each scalar c and every u in H, the scalar multiple cu is in H
let H be a subset of a vector space V. A set of vectors B in V is a basis for H if:
1) B is a linearly independent set
and
20 H = span {B}
what is the spanning set theorem?
a basis is an “efficient” spanning set that contains no unnecessary vectors, a spanning set is a set of vectors that are linearly independent and span the space (definition of basis)
the pivot columns of a matrix A form ________
a basis for Col A
if two matrices A and B are equivalent, then their _______
row spaces are the same
if a vector space V has a basis of exactly n vectors, then every basis for V _________
must consist of exactly n vectors
