Alg Flashcards
A set of vectors {e1, e2, . . . , ed} is called a ____ if every vector v can be uniquely written as a weighted sum or linear combination of the basis vectors according to v = Σ(i=1 to d) viei.
The coefficients v1, v2, . . . , vd are called the vector coordinates in that basis (Like when we have 2[0,1]+3[1,0] which forms v=[2,3] and the coordinates are x=3 and y=2).
P9
basis
- In one dimension, a single non-zero vector forms a basis;
- in two dimensions, two vectors that do not lie on the same line form a basis
- in three dimensions, three vectors that do not lie in the same plane form a basis
- a basis must contain as many vectors as there are dimensions
If all basis vectors have unit magnitude ||ei|| = 1 and are aligned with the axes of the (Cartesian)coordinate system, then they form a ____ basis or canonical basis
P10
Standard
A general such system can be solved by means of Gaussian elimination (usual method of solving n equations n variables)
An upper-triangular matrix contains ____ below the main diagonal; a lower-triangular matrix contains ____ above the main diagonal. A diagonal matrix only contains ____ elements on the main diagonal.
zeros
zeros
non-zero
What’s an orthogonal matrix?
The inverse of an orthogonal matrix is its transpose:
𝑄−1=𝑄𝑇
For an orthogonal matrix all columns as well as all rows form an orthonormal basis; the standard basis is a special case of an orthonormal basis.
Calculating a 3 x 3 matrix inverse
Another easier method for 3 x 3 matrices
Note: adjoint matrix is the transpose of cofactor matrix
Note: Calculate the Determinant of Matrix using the first row,
Det A = 2(cofactor of 2) + 1(cofactor of 1) + 3(cofactor of 3)
Note: GeekforGeek method is true for all matrices.
Left-Inverse and Right-inverse matrices
Pseudo-inverse matrix
Non-square matrices, i.e. m-by-n matrices for which m ≠ n, do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If A is m-by-n and the rank of A is equal to n, (n ≤ m), then A has a left inverse, an n-by-m matrix B such that BA = In. If A has rank m (m ≤ n), then it has a right inverse, an n-by-m matrix B such that AB = Im
The determinant of any square matrix A is a scalar, denoted det(A). [Non-square matrices do not have determinants.]
The determinant of a lower-triangular, upper-triangular, or diagonal matrix equals the ____.
P32
product of the main diagonal elements
What’s the null space (or kernel) of a matrix 𝐴?
P35
Answer
Exp
Khan Academy video on null and column spaces
Note: Pivot entries. All columns in RREF with only one 1 in them, are pivot columns and to find the null space, we should solve for these columns.
REF: Simplifies systems of linear equations but does not uniquely determine a form.
RREF: Provides a unique form, making it more powerful for solving systems of equations and understanding the structure of the linear system.
The rank of matrix A is ____
How is the row space of a matrix calculated?
What is the left null space of matrix A?
dimensionality of the left null space, i.e. the number of basis vectors that spans the left null space, is called the ____ of the matrix A.
Rank A =? Rank A T
The dimensionality of the row space, i.e. the number of basis vectors that spans the row space, is called the row rank of the matrix A.
The dimensionality of the column space, i.e. the number of basis vectors that spans the column space, is called the column rank of the matrix A.
The rank-nullity theorem states that for nxd matrix A ____
P36
rank(A)+nullity(A)=d. By applying the same rule to
AT we find that rank(A)+corank(A)=n
The left and right null spaces and the row and column spaces of an nxd matrix A are collectively referred to as the four ____ of a matrix.
P35
fundamental subspaces
If a system Ax=0 can be reduced by means of Gaussian elimination to a row echelon form Rx=0, then the non-zero rows of R form a basis for the ____ (row/column) space of A; the columns of A that correspond with the pivot elements in R form a basis for the ____ (row/column) space of A.
P36
row
column
A mapping is surjective if ____
and injective if ____;
if ____, the mapping is bijective. A linear bijective mapping is characterised by an invertible matrix.
P40
for all y there is an x that projects to it
different x always project to different y (not covering the whole range)
both hold
schema
P40
