Linear Algebra Flashcards

1
Q

What does linear algebra is all about

A

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

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2
Q

What is a System of Linear Equations

A

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

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3
Q

What is a solution to a system of linear equations

A

solutions correspond to how lines can meet

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4
Q

What are matrices used for

A

to understand and perform operations on large number of lines

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5
Q

What are the three posssible number of solutions to a system

A

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.

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6
Q

What are the two different types of matrices

A

Coefficient matrix
Augmented matrix

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7
Q

What does augmented matrix contain

A

an extra column for the values after equal to

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8
Q

When do you call a system consistent

A

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

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9
Q

What is the condition to call a matrix a echelon matrix

A

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.

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10
Q

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

A

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.

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11
Q

What is row reduction / Gaussian elimination

A

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

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12
Q

what is pivot position and pivot position

A

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.

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13
Q

what is the relation between vector and matirx

A

Vector is a single row in a matrix

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14
Q

what is a vector space

A

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

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15
Q

What does a span mean

A

Span indicates subset of vector space which is of intrest

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16
Q

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

A

this expands the vector

17
Q

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

A

we are compressing the vector

18
Q

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

A

it changes the direction of the vector

19
Q

What is the basis vector

A

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

20
Q

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

A

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

21
Q

What is linear independence

A

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

22
Q

What does it mean when the vectors has a trivial solution

A

This means that the vectior is linear independenta

23
Q

What does it mean a vector is linear dependent

A

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

24
Q

how to find linear dependence using p and n values

A

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

25
what does p and n in a vector
A - { [1] [3] [4] } [2] [4] [2] here number p ie columns is 3, but the n ie rows is 2
26
what is dot product multiplication of matrices
if the multiplication of a vector result in a scalar its called as dot product multiplication of matrices or scalar multiplicaiton of matrices
27
what is vector multiplication of matrices
if the multiplication of a vector result in a vector its called as vector multiplication of matrices
28
What does Image. Preimage mean inrelation to linear transfomration
preimage - input image - output
29
how do we correlate domain and range in relation to linear transformation
range is the all possible images, domain is all possible preimages
30
What are conditions to call a transfomation linear
vector addition and scalar multiplication T(u+v) = Tu +Tv Scalar multiplication T(cv) = Tc * v
31
What does Linear transformation do to a vector
Linear Transformation either stretches, compresss, flip or rotate the vector
32
in Linear transfomation of a vector what does not change
the origin of the vector doesnot change
33
can transformations can be applied to entrie vector space or is it limited to the vector
Transformations can ber applied to the entire vector space
34
Relation between Linear tranformation and matrix
Linear transformation is nothing but vector multiplication in respect to matrices
35
Which transformation is called as Algebric and which is geomentric
Linear trans is algerbric, vector trans is geometric
36
What is onto in linear transformation
if every range equates to every domain then its onto
37
What is one onto in linear transformation
if every one range equates to every one domain then its one onto
38
What is not onto
if every range equates to only one domain then its not onto
39