Midterm 1 Flashcards
Equation of a line in a plane
ax + by = c
Compatible
equation = # variables
1 solution
# equations = # variables
Incompatible
equations = # variables
0 solution
[0 0 0 … 1]
Compatible indeterminate
equations < # variables
infinite solutions
# equations < # variables
Elementary row operations
don’t change solution set
1. Interchanging two rows
2. Multiplying a row by a non-zero constant
3. Adding to a row a multiple of another
Vector
(m x 1) matrix
RREF rules
- Leading 1s are the first # unless all 0 row
- Row with only 0s at the bottom
- Lower leading one must be farther to the right
- Each leading 1 column has 0s everywhere else in the column
Is matrix addition associative?
Yes
(A+B)+C=A+(B+C)
Is matrix addition commutative?
Yes
A+B=B+A
What is the transpose of a matrix?
Interchange rows & columns
What does it mean for a matrix to be symmetric?
Square matrix is equal to its transpose
Span
set of vectors you can reach with a linear combinations
(A^T)T =
(A^T)T = A
What are three ways to solve a linear system of equations?
- Equation: x1v1 + … + xnvn = b
- System: Ax = b
- Augmented matrix: [a1 … an | b]
Linearly dependent:
if there exists scalars (not all equal to zero) such that:
a1v1 + a2v2 + … + anvn = 0
Linearly independent:
The only scalars such that a1v1 + a2v2 + … + anvn = 0 are all equal to zero
If a set of equations has more variables than equations, then is it linearly independent or dependent?
linearly dependent
Is the zero vector linearly indpendent or dependent?
linearly dependent
Given a matrix in REF, the non-zero rows are
linearly independent
If two vectors are linearly independent are you perform elementary row operations on them, are they still independent?
Yes
Rank
linearly independent rows of the matrix
leading 1s in the general solution of Ax=0 (REF)
If A and B are matrices,
Does AB = BA?
NO
(AB)^T =
(AB)^T = B^T A^T
Homogeneous linear system
Ax = 0
Solution set sends vectors to zero vector
Non-homogeneous linear system
Ax = b
Solutions to Ax=b are translations of the solutions to Ax=0
If a is a solution to Ax = 0 and k is scalar, what is another solution to Ax = 0?
ka
If a1 and a2 are solutions to Ax=0, then what is another solution to Ax=0?
a1 + a2
More generally iff a1 and a2 are solutions to Ax=0 and k are scalars, then what?
Any linear combination (k1a1+…knan) is also a solution to Ax = 0
If a is a solution to Ax=0 and c is a solution to Ax=b, then what is another solution to Ax=b?
c+a solution to Ax=b
If c1 and c2 are solutions to Ax=b, then what is a solution to Ax=0?
c1 - c2 is a solution to Ax=0
What is a linear transformation? T: Rn -> Rm such that
- T(v + w) = T(v) + T(w)
- T(kv) = kT(v)
Any matrix transformation can be written as what?
A unique matrix times a vector
T(x) = Ax
If T is a linear transformation then, T(0) =
T(0) = 0
What is another way of writing the linear transformation T(x)?
T(x) = Ax
What is the matrix of rotation by angle theta?
Invertible Matrix
Square matrix such that:
AA^-1 = I and A^-1A = I
What is the condition that tells us if a square matrix is invertible?
det(a) ≠ 0
otherwise space would be squashed into a lower dimension and there is no inverse transformation that could undo that
How can you find the inverse of a matrix?
[ A | I ] –> RREF [I | A^1]
Augment it with the identity and do row operations
For a square matrix, what statements are equivalent regarding inverses?
- There exists a matrix B such that BA = I
- There exists a matrix C such that AC = I
- The matrix A has rank n (full)
- For every vector b, the equation Ax=b has a unique solution
Can you find the inverse of an (M x N) matrix
Only have an inverse on one side
Left inverse: iff it has linearly independent columns
Right inverse: iff it has linearly independent rows
Vector space
A set of vectors V which has:
- u + v in V
- ku in V
Must satisfy 8 axioms
What are the 8 axioms of a vector space?
- u + v = v + u
- u + (v + w) = (u + v) + w
- u + 0 = u
- u + (-u) = 0
- (k1k2)u = k1(k2u)
- (k1 + k2)u = k1u + k2u
- k(u + v) = ku + kv
- 1u = u
