Flashcard
(25 cards)
The span of a set of vectors is the set of all ________ ________ of the vectors. In R3, the span must either be ________, ________, ________, or ________.
linear combination, 0, line through origin, plane through origin, all of R3
If S is a subspace of a vector space V , then the following four properties must hold:
- Closed under addition
- Closed under scalar multiplication
- Contains zero vector
- If x is in S, then it is in V
A linear combination of the vectors {v1, v2,…, vk} is defined as:
x1v1 + x2v2 +…+ xnvn = b
If a set of vectors is linearly independent, then no vector is a ________ ________ of the other vectors in the set. To determine the number of linearly independent vectors, we row reduce and count the number of ________ columns.
linear combination, pivotal
If T: Rn –> Rm is a linear transformation, then the matrix A of T has dimension ________. The domain of T is ________ and the codomain of T is ________.
m x n, Rn, Rm
Finding the range of a transformation is equivalent to finding the ________ of the columns of A.
span
If a linear transformation is 1-to-1, then the row-reduced matrix must have a pivotal one in every ________. If the transformation is onto, then the row-reduced matrix has a pivotal one in every ________.
column, row
The RREF form of A is _________.
In
The equation Ax = b always has a ________ solution.
unique
The columns of A form a ________ ________ set.
linearly independent
The columns of A ________ Rn.
span
The only solution to Ax = 0 is ________.
x = 0
The transformation TA: Rn –> Rn, represented by T(x) = Ax, is ________ and ________.
onto, one-to-one
The range of TA is ________.
Rn
The determinant of A is ________.
nonzero
If the set of vectors B forms a basis for the vector space V , then all vectors in B are ________ ________ and the span of B is ________.
linearly independent, V
The dimension of a vector space is defined as the number of vectors in a ________ for the space.
basis
If the dimension of V is n, then any set of more than n vectors will not be ________, and any set of less than n vectors will not ________ ________.
linearly independent, span V
To find the rank of a matrix, we count the number of ________ columns.
pivotal
If A is an m x n matrix, then the equation relating rank to nullity is: ________.
rank + nullity
To find a basis for the ________ space of A, use the pivotal columns from A.
column
To find a basis for the ________ space of A, solve the equation Ax = 0.
null
Write out two other words for the column space: ________ or ________.
range, span
Write out another word for the null space: ________.
kernal
