1 | review matrix, vector spaces Flashcards
m x n ?
m rows
n columns
Additive identity of a matrix
Zero matrix of the same order is an additive identity
A + 0 = 0 + A = A
Additive inverse?
βπ΄ is the additive inverse, since
π΄ + (βπ΄) = (βπ΄) + π΄ = 0
If π΄ is an π Γ π matrix and π΅ is an π Γ π matrix,
the product πΆ is an _? x _? matrix whose entries are given by _______?
What is this product called?
m x p
cij = ai. b.j
dot product
orthogonal vectors?
example?
Two vectors are said to be orthogonal (perpendicular) if their dot product is zero.
Example. π = 1 2 3 , π = β2; 1; 0
π β π = 0
Is matrix multiplication (dot product) commutative?
no
Square matrix - definitiion?
diagonal of a square matrix?
A square matrix is a matrix which has the same number of rows and columns.
The diagonal of a square matrix is given by the entries which have the same row and column indices.
Diagonal matrix?
A diagonal matrix is a square matrix whose all off-diagonal entries are zero
Identity matrix?
Operations?
The identity matrix πΌπ is a diagonal matrix whose all diagonal entries are 1.
For an π Γ π matrix π΄, it holds that πΌmπ΄ = π΄ and π΄πΌπ = π΄
Examples.
For an π Γ π matrix π΄, it holds that ____π΄* = π΄
πΌπ*π΄
For an π Γ π matrix π΄, it holds that π΄*____ = π΄ ?
π΄*πΌπ
Matrix operation:
commutative law?
only for addition!
π΄ + π΅ = π΅ + π΄
Matrix operatiom:
Scalar multiplication law?
(ππ΄)π΅ = π΄(ππ΅) = π(π΄π΅)
Matrix operation:
Distributive law?
(π΄ + π΅)πΆ = π΄πΆ + π΅πΆ
π΄(π΅ + πΆ) = π΄π΅ + π΄πΆ
Matrix operation:
Associative law?
(π΄ + π΅) + πΆ = π΄ + (π΅ + πΆ)
(π΄π΅)πΆ = π΄(π΅πΆ)
Transpose matrix definition
If π΄ is an π Γ π matrix,
the transpose πΆ = π΄π is an π Γ π matrix
whose entries are given by πππ = πππ
In other words, the π row of π΄ becomes the π column in π΄π
Symmetric Matrix?
Matrix is symmetric if π΄π = π΄
Assuming that all products and sums below are defined, the
following hold:
(π΄ + π΅)π = ?
(π΄π΅)π = ?
(π΄ + π΅)π = Aπ + Bπ
(π΄π΅)π = BπAπ
Homogeneous system of linear:
Definition?
How can it be solved?
π΄π₯ = π
If all entries in π are zero, the system is homogeneous
Otherwise, non-homogeneous
can be solved by Gaussian elimination
Row echelon form
Definition?
A matrix is said to be in a row echelon form if
(i) The first non-zero entry in every row is to the right of the first non-zero entry in all the rows above
(ii) Every entry above a first non-zero entry is zero
Row echelon form:
pivot?
Pivot (corner entry) = the first non-zero entry in a row
Row Echelon Form:
reduced?
Reduced row echelon form = every pivot equals one!
The reduced form of matrix π΄ is denoted by π΄πππ
Elementary row operations on a matrix?
Use?
Three elementary row operations on a matrix π΄ are defined as follows:
- Interchange two rows of π΄
- Multiply a row of π΄ by a non-zero scalar (i.e. constant)
- Replace a row of π΄ by itself plus a multiple of a different row
Every matrix can be put into a (unique) reduced row echelon form by applying a (not unique) sequence of elementary row operations.
Matrix rank
The reduced form of matrix π΄ is denoted by π΄πππ
The row rank, or simply, the rank, of a matrix π΄ is the number of non-zero rows in π΄πππ