midterm 2 Flashcards
let u, v, and w be vectors in R^n, and let c be a scalar. Then, _________________ (properties of the inner product)
u * v = v * u
(u + v) * w = u * w + v * w
(cu) * v = c(u * v) = u * (cv)
u * u >= 0 +, and u * u = 0 in and only if u = 0
the length (or norm) of v is the nonnegative scalar ||v|| defined by ______________
||v|| = √(v * v) = √(v1^2 + v2^2 + … Vn^2)
and
||v^2|| = v * v
for u and v in R^n, the distance between u and v, written as dist(u, v) is the length of the vector u - v. That is, ______________
dist(u, v) = ||u - v||
two vectors u and v in R^n are orthogonal (to each other) if __________
u * v = 0
two vectors u and v are orthogonal if and only if __________________
||u + v||^2 = ||u^2|| + ||v^2||
how do you check to see if a set of vectors is an orthogonal set?
check the dot product between each of the vectors
if S is an orthogonal set of nonzero vectors in R^n, then S is _________________
linearly independent and a basis for the subspace spanned by S
an orthogonal basis for a subspace W of R^n is a _________________
basis for W that is also an orthogonal set
let {u1 ….. up} be an orthogonal basis for a subspace W or R^n. For each y in W, the weights in the linear combination y = c1u1 + … + cpup are given by ________________
cj = (y * uj) / (uj * uj)
a set is an orthonormal set if ________________
it is an orthogonal set of unit vectors
how do we check if a set is orthonormal?
1) check if the set is orthogonal
2) check if the lengths of each of the vectors is 1
an mxn matrix U has orthonormal columns if and only if ________
U^T * U = I
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 vector space axioms?
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
what are the properties to be a subspace of a vector space?
1) the zero vector of V is in H
2) H is closed under vector addition. This means for each vector u and v in H, the sum u + v is in H
3) H is closed under scalar multiplication. This means for each vector u in H and every scalar c, the vector cu is in H
the null space of an mxn matrix A, written as Nul A, is the _______________________
set of all solutions to the homogeneous equation Ax = 0.
what is nullspace in set notation?
Nul A = {x : x is in R^n and Ax = 0}
the column space of an mxn matrix A, written as Col A, is _________________
the set of all linear combinations of the columns of A
If A = [a1 … an], the column space of A is ________
Col A = Span{a1, … an}
or in set notation
Col A = {b : b = Ax for some x in R^n}
the row space of an mxn matrix A, written as Row A, is __________________
the set of all linear combinations of the row vectors in A
if v1 … vp are in a vector space V, then _______
Span{v1 … vp} is a subspace of V
the set of all solutions to the system Ax = 0 of m homogeneous linear equations and n unknowns (null space) is _______________
a subspace of R^n
let H be a subspace of a vector space V. A set of vectors B in V is a basis for H if ______
i) B is a linearly independent set
ii) the subspace spanned by B coincides with H, meaning H = Span B
if A is an invertible matrix,
the columns of A form a basis for R^n because they are a linearly independent set and span R^n
the pivot columns of a matrix A _________
form a basis for Col A
if two matrices are row equivalent, __________
then their row spaces are the same
if two matrices are row equivalent and B is in echelon form, ___________
the nonzero rows of B form a basis for the row space of A and B
if a vector space V has a basis B = {b1, … b2}, then any set in V __________________
containing more than n vectors must be linearly dependent
if a vector space V has a basis of n vectors, ____________
then every basis of V must consist of exactly n vectors
if a vector space V is spanned by a finite set, __________________________
then V is said to be finite-dimensional and the dimension of V, written as dim V, is the number of vectors in a basis for V.
let V be a p-dimensional vector space.
any linearly independent set of exactly p elements in V is automatically a basis for V.
any set of exactly p elements that spans V is automatically a basis for V
