Alg Flashcards
(36 cards)
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.
