01 Linear Equations in Linear Algebra Flashcards
consistent system of linear equations
A system of linear equations is said to be consistent if it has either one solution or infinitely many solutions; a system is inconsistent if it has no solution.
Elementary row operations
- Replacement
- Interchange
- Scaling
echelon form
A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties:
- All nonzero rows are above any rows of all zeros
- Each leading entry of a row is in a column to the right of the leading entry of the row above it
- All entries in a column below a leading entry are zeros
If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form:
- The leading entry in each nonzero row is 1
- Each leading 1 is the only nonzero entry in its column
Each matrix is row equivalent to one and only one _____
Each matrix is row equivalent to one and only one reduced echelon matrix
RREF
Reduced Row Echelon Form
pivot position, pivot column
A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is column of A that contains a pivot position.
RREF places each basic variable in ___ equation
RREF places each basic variable in one and only one equation
The variables corresponding to pivot columns in the matrix are called ____. The other variables are called ____.
The variables corresponding to pivot columns in the matrix are called basic variables. The other variables are called free variables.
Row Reduction Algorithm
…
The solution is not unique because there are ___ variables.
The solution is not unique because there are free variables.
Existence and uniqueness theorem
A linear system is consistent if and only if the rightmost column of the augmented matrix not a pivot column.
Span
If v1, …, vp are in Rn, then the set of all linear combinations of v1, …, vp is denoted by Span{v1, …, vp} and is called the subset of Rn spanned by v1, …, vp.
Asking whether a vector b is in Span {v1, …, vp} amounts to asking whether_____
Asking whether a vector b is in Span {v1, …, vp} amounts to asking whether the linear system with augmented matrix [v1 … vp b] has a solution.
The equation Ax = b has a solution if and only if ___
The equation Ax = b has a solution if and only if b is a linear combination of the columns of A.
Let A be and m x n matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false:
a. For each b in Rm, the equation Ax = b has a solution
b. Each b in Rm is a linear combination of the columns of A
c. The columns of A span Rm.
d. A has a pivot position in every row.
A system of linear equations is said to be homogeneous if it can be written in the form ____
Ax = 0.
For a homogeneous system of linear equations, the important question is whether there exists a ___
For a homogeneous system of linear equations, the important question is whether there exists a nontrivial solution, that is, a nonzero vector x that satisfies Ax = 0.
The homogeneous equation Ax = 0 has a nontrivial solution if and only if ____
The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.
Suppose the equation Ax = b is consistent for some given b, and let p be a solution. Then the solution set of AX = b is the set of all vectors of the form w = p + v_h, where v_h is any solution of ___
Suppose the equation Ax = b is consistent for some given b, and let p be a solution. Then the solution set of AX = b is the set of all vectors of the form w = p + v_h, where v_h is any solution of the homogeneous equation Ax = 0.
An indexed set of vectors {v1, …, vp} in Rn is said to be linearly independent if ____
An indexed set of vectors {v1, …, vp} in Rn is said to be linearly independent if the vector equation
x1vi + x2vi + … + xpvp = 0
has only the trivial solution.
The ___ of a matrix A are linearly independent if and only if the equation ___ has only the trivial solution
The columns of a matrix A are linearly independent if and only if the equation Ax = 0 has only the trivial solution
Characterization of Linearly Dependent Sets
An indexed set S = {v1, …, vp} of two or more vectors is linearly dependent if and only if at least one of the vectors in S is linear combination of the others. In fact, if S is linearly dependent and v1 != 0, then some vj (with j > 1) is a linear combination of the preceding vectors, v1, …, v_{j-1}.
Every vector in a linearly dependent set ___ a linear combination of the other vectors.
is not necessarily
If a set contains more vectors than there are entries in each vector, then the set is _____.
If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v1, …, vp} in Rn is linearly dependent if p > n.
If a set S = {v1, …, vp} in Rn contains the zero vector, then the set is _____.
If a set S = {v1, …, vp} in Rn contains the zero vector, then the set is linearly dependent.
A transformation T from Rn to Rm is a rule that assigns to each vector x in RN A VECTOR T(x) in Rm. The set Rn is called the ____ of T, and Rm is called the ____ of T. For x in Rn, the vector T(x) in Rm is called the ____ of x (under the action of T). The set off all images T(x) is called the ____ of T.
A transformation T from Rn to Rm is a rule that assigns to each vector x in RN A VECTOR T(x) in Rm. The set Rn is called the domain of T, and Rm is called the codomain of T. For x in Rn, the vector T(x) in Rm is called the image of x (under the action of T). The set off all images T(x) is called the range of T.
A transformation T is linear if:
A transformation T is linear if:
- T(u + v) = T(u) + T(v) for all u, v in the domain of T
- T(cu) = cT(u) for all scalars c and all u in the domain of T
Let T : Rn -> Rm be a linear transformation. Then there exists a unique matrix A such that
T(x) = _____
In fact, A is the m x n matrix whose jth column is the vector T(e_j), where e_j is the jth column of the identity matrix in Rn:
A = _____
Let T : Rn -> Rm be a linear transformation. Then there exists a unique matrix A such that
T(x) = Ax for all x in Rn
In fact, A is the m x n matrix whose jth column is the vector T(e_j), where e_j is the jth column of the identity matrix in Rn:
A = [T(e1) … T(en)]
The matrix A is called the standard matrix for the linear transformation T.
A mapping T : Rn -> Rm is said to be onto Rm if ____
A mapping T : Rn -> Rm is said to be onto Rm if each b in Rm is the image of at least on x in Rn (= when the range of T is all of the codomain Rm).
A mapping T : Rn -> Rm is said to be one-to-one if _____
A mapping T : Rn -> Rm is said to be one-to-one if each b in Rm is the image of at most one x in Rn.
T is one-to-one if and only if ____
T is one-to-one if and only if T(x) = 0 has only the trivial solution
T maps Rn onto Rm if and only if ____
T maps Rn onto Rm if and only if the columns of A span Rm
