Linear Algebra

Question 1

What does linear algebra is all about

Answer 1

Linear Algebra helps us to understand mathematics and science behind two or more lines

Question 2

What is a System of Linear Equations

Answer 2

It is a collection of one or more linear equations that use the same variables.

Question 3

What is a solution to a system of linear equations

Answer 3

solutions correspond to how lines can meet

Question 4

What are matrices used for

Answer 4

to understand and perform operations on large number of lines

Question 5

What are the three posssible number of solutions to a system

Answer 5

There are three possibilities for a system of linear equations.

The system has no solutions.

The system has exactly one solution.

The system has infinitely many solutions.

Question 6

What are the two different types of matrices

Answer 6

Coefficient matrix
Augmented matrix

Question 7

What does augmented matrix contain

Answer 7

an extra column for the values after equal to

Question 8

When do you call a system consistent

Answer 8

A consistent system of linear equations has at least one solution. An inconsistent system of linear equations has no solutions.

Question 9

What is the condition to call a matrix a echelon matrix

Answer 9

3 conditions
1) Any rows consisting of all 0’s are below all nonzero rows.
2)Each leading entry of a row is in a column to the right of the leading entry of any row above it. (steps)
3)All entries in a column below a leading entry are 0.

Question 10

What is the condition to call a matrix a reduced row echelon matrix

Answer 10

all the three conditions on echelon matrix and
4) The leading entry of each nonzero row is 1
5) Every leading 1 is the only nonzero entry in its column.

Question 11

What is row reduction / Gaussian elimination

Answer 11

Row reduction is the process of transforming a matrix into echelon form with elementary row operations. Row reduction is also called Gaussian elimination.

Question 12

what is pivot position and pivot position

Answer 12

A pivot position is a position in a matrix that corresponds to a leading 1 in the matrix’s reduced echelon form

A pivot column is a column in a matrix that contains a pivot position.

Question 13

what is the relation between vector and matirx

Answer 13

Vector is a single row in a matrix

Question 14

what is a vector space

Answer 14

vector space is more like a population in statistics, it entitles all the vectors in the given dimension

Question 15

What does a span mean

Answer 15

Span indicates subset of vector space which is of intrest

Question 16

What happens when you multiply a vector with a scalar value more than one

Answer 16

this expands the vector

Question 17

What happens when you multiply a vector with a scalar value between 0 and 1

Answer 17

we are compressing the vector

Question 18

What happens when you multiply a vector with a scalar value less than 1

Answer 18

it changes the direction of the vector

Question 19

What is the basis vector

Answer 19

Suppose we are in a R2 vector space(Two dimensional vector),
the axis x and y which coresponds to the vector is called as basis vector

Question 20

What is the difference between vector and matrices in relation to multiplication

Answer 20

Vector multiplication is commutative, but matirx multiplication is not commutative
ie, a vector * b vector = b vector*a vector but
A matrix * B matrix is not equal to B matrix *A matrix

Question 21

What is linear independence

Answer 21

A set of vector is set to be linear independence if sum of vectors equals to zero,
this means it has a trivial solutioin

Question 22

What does it mean when the vectors has a trivial solution

Answer 22

This means that the vectior is linear independenta

Question 23

What does it mean a vector is linear dependent

Answer 23

it means any of the vectors in the vector space is not equal to zero,
we can equate one vector to another by moving sides

Question 24

how to find linear dependence using p and n values

Answer 24

if p >n then its linear dependent, else we need to check

Question 25

what does p and n in a vector

Answer 25

A - { [1] [3] [4] }
[2] [4] [2]
here number p ie columns is 3,
but the n ie rows is 2

Question 26

what is dot product multiplication of matrices

Answer 26

if the multiplication of a vector result in a scalar its called as dot product multiplication of matrices or scalar multiplicaiton of matrices

Question 27

what is vector multiplication of matrices

Answer 27

if the multiplication of a vector result in a vector its called as vector multiplication of matrices

Question 28

What does Image. Preimage mean inrelation to linear transfomration

Answer 28

preimage - input
image - output

Question 29

how do we correlate domain and range in relation to linear transformation

Answer 29

range is the all possible images, domain is all possible preimages

Question 30

What are conditions to call a transfomation linear

Answer 30

vector addition and scalar multiplication
T(u+v) = Tu +Tv

Scalar multiplication
T(cv) = Tc * v

Question 31

What does Linear transformation do to a vector

Answer 31

Linear Transformation either stretches, compresss, flip or rotate the vector

Question 32

in Linear transfomation of a vector what does not change

Answer 32

the origin of the vector doesnot change

Question 33

can transformations can be applied to entrie vector space or is it limited to the vector

Answer 33

Transformations can ber applied to the entire vector space

Question 34

Relation between Linear tranformation and matrix

Answer 34

Linear transformation is nothing but vector multiplication in respect to matrices

Question 35

Which transformation is called as Algebric and which is geomentric

Answer 35

Linear trans is algerbric, vector trans is geometric

Question 36

What is onto in linear transformation

Answer 36

if every range equates to every domain then its onto

Question 37

What is one onto in linear transformation

Answer 37

if every one range equates to every one domain then its one onto

Question 38

What is not onto

Answer 38

if every range equates to only one domain then its not onto