Alg

Question 1

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

Answer 1

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

Question 2

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

Answer 2

Standard

Question 3

How is dot product of two vectors calculated?

P13

Answer 3

Answer

Question 4

What is orthonormal basis?

P13

Answer 4

Answer

Question 5

How is the dot product calculated for two vectors with orthonormal basis?

P13

Answer 5

Answer

Question 6

Identities of dot product

Question 7

homogeneous linear systems

A general such system can be solved by means of Gaussian elimination (usual method of solving n equations n variables)

Question 8

When are a set of vectors linearly dependent?

Answer 8

Answer

Question 9

When does a set of vectors, span the entire R^d space?

P19

Answer 9

Answer

Question 10

How is the norm of a matrix calculated?

P23

Answer 10

Answer

Question 11

What are transpose, symmetric and Asymmetric matrices?

Answer 11

Answer

Question 12

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.

Answer 12

zeros
zeros
non-zero

Question 13

Trace

Note that trace is defined for square matrices

Question 14

How is a 2 * 2 matrix’s inverse calculated?

P25

Answer 14

Answer

Question 15

What’s an orthogonal matrix?

Answer 15

Answer

The inverse of an orthogonal matrix is its transpose:
𝑄⁻¹=𝑄^𝑇

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.

Question 16

Matrix Operations

Question 17

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.

Question 18

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 = I_n. If A has rank m (m ≤ n), then it has a right inverse, an n-by-m matrix B such that AB = I_m

Question 19

Determinant properties

The determinant of any square matrix A is a scalar, denoted det(A). [Non-square matrices do not have determinants.]

Question 20

The determinant of a lower-triangular, upper-triangular, or diagonal matrix equals the ____.

P32

Answer 20

product of the main diagonal elements

Question 21

Solving Equations using the cramer’s rule

Question 22

What’s the null space (or kernel) of a matrix 𝐴?

P35

Answer 22

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.

Question 23

Row Echelon Form

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.

Question 24

Basis for null space and nullity

Question 25

How do you know if the column vectors of a matrix are linearly independent?

Question 26

A matrix’s column space is the span of its columns, the span’s basis are the columns associated with the pivot columns of the RREF of the matrix

Question 27

The rank of matrix A is ____

Answer 27

The number of vectors in the basis for the column space

Question 28

How is the row space of a matrix calculated?

Answer 28

The column space of the matrix’s transpose

Question 29

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.

Answer 29

The null space of the A’s transpose
corank

Question 30

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.

Answer 30

it’s equal

Question 31

The rank-nullity theorem states that for nxd matrix A ____

P36

Answer 31

rank(A)+nullity(A)=d. By applying the same rule to
A^T we find that rank(A)+corank(A)=n

Question 32

The columns of the transformation matrix A (nxd) are the images of the basis vectors e1,e2,..,ed respectively.
Exp

Question 33

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

Answer 33

fundamental subspaces

Question 34

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

Answer 34

row
column

Question 35

A mapping is surjective if ____
and injective if ____;
if ____, the mapping is bijective. A linear bijective mapping is characterised by an invertible matrix.

P40

Answer 35

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

Question 36

Linear Mapping

P40