Exam 2 Flashcards
Testing for row equivalence
Every matrix row equivalence class reduces to the same rref matrix.
Showing row operations to relate two matrices as row-equivalent
A = (α1, α2…)
Show that B = (β1=α1, β2=2α2… etc)
Echelon form definition
No non-zero row is a linear combination of any of the other rows.
Verify:
row1 /= (row2)c1 + (row3)c2 …+ (rowN)cN
A system of equations has ___ matrices associated with it.
3:
Coefficient matrix
Homogeneous matrix
Non-homogeneous matrix
Equivalent statements for:
A is nonsingular
A is nonsingular.
rref [A] = identity matrix.
The homogeneous system has one solution: the zero vector.
T/F: All nonsingular matrices are in the same row-equivalence class.
True.
Nonsingular means rref[A]= identity matrix
T/F: All NxN (square) matrices are in the same row equivalence class.
False
If two systems (“systems” includes “=d”, nonhomogeneous) have row equivalent matrices, do they have the same solution?
Yes
How many members can there be in a row equivalence class?
Either one (the zero matrix) or infinite.
Rules to determine if a set is a vector space.
Addition: u+v ε V
Scalar multiplication: rv ε V
Zero vector: 0 ε V
The improper subspaces of V
V itself and the trivial subspace (0)
Constructing a subset
Put some restriction on the vector space.
I.e. “such that…”
Constructing a subset that is a subspace of the vector space.
Put a restriction on the vector space “such that…” which abides by the rules of a space (addition, scalar multiplication, zero vector).
Equivalent statements:
S is a subspace of V
S is a subspace of V.
S is “closed” (valid) under linear combinations of any N of its vectors.
S is a subspace of V. Find all vectors that can be used in equivalent statements.
Parameterize the statement (free variables times vectors) so that S is the set of all linear combinations of those vectors
What is a linear combination?
Coefficients times the vectors of a system
What is span?
The span of a subset S is:
the set of all linear combinations of vectors within S
Note: [S] is a vector space, and will be in the form of a line (one parameter), plane (two parameters) etc
Test for linear independence
Set S is linearly independent iff
c1s1 + … + cNsN = 0
has ONLY the trivial solution:
c1 = c2 = … = cN = 0
Note: S is linearly dependent if the restriction is c1 = c2 = etc… Or if the zero vector is included in the set
How to identify/remove the unnecessary vectors in a linearly dependent set? (How to create a subset with the same span)
Vectors times coefficients (c1, c2, etc) = 0.
Convert to a system or matrix, solve, discard any of the original vectors that are attached to the resulting free variables
Minimal =
Linearly independent
Maximal =
Spans the space
Linearly dependent subset of a linearly dependent set S:
Does not exist
Linearly independent subset of a linearly independent set S:
S, the original set itself
Define: basis
A minimal and maximal set
linearly independent and spans the space
Does it span the space?
= “Do c1, c2, etc… exist such that you can produce all the vectors in the space?”
For a matrix: transpose, reduce to rref, number of nonzero rows must match # of dimensions in a space if it spans (bc that’s the # of linearly independent vectors in the set)
“Representation of vector v with respect to basis B”
RepBv =
RepBv = ( c1 ) ( c2 ) ( ... ) ( cN )
Where v = c1b1 +…+ cNbN
(b’s = members of basis B)
Find span(S) in V.
Parameterize the set (c1, c2, etc) and
set equal to the span of V
(like (x, y, z) or (a0 + a1x + a2x^2),
Put into a system of eq’s and solve. Right side of eq’s are either unrestrained (then [S]=V) or restrained by some combo of parameters (a’s or xyz’s). Put parameters with the span of V (like R3 vector or P2 polynomial) to get the span of S
