Linear Algebra Flashcards
What does linear algebra is all about
Linear Algebra helps us to understand mathematics and science behind two or more lines
What is a System of Linear Equations
It is a collection of one or more linear equations that use the same variables.
What is a solution to a system of linear equations
solutions correspond to how lines can meet
What are matrices used for
to understand and perform operations on large number of lines
What are the three posssible number of solutions to a system
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.
What are the two different types of matrices
Coefficient matrix
Augmented matrix
What does augmented matrix contain
an extra column for the values after equal to
When do you call a system consistent
A consistent system of linear equations has at least one solution. An inconsistent system of linear equations has no solutions.
What is the condition to call a matrix a echelon matrix
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.
What is the condition to call a matrix a reduced row echelon matrix
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.
What is row reduction / Gaussian elimination
Row reduction is the process of transforming a matrix into echelon form with elementary row operations. Row reduction is also called Gaussian elimination.
what is pivot position and pivot position
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.
what is the relation between vector and matirx
Vector is a single row in a matrix
what is a vector space
vector space is more like a population in statistics, it entitles all the vectors in the given dimension
What does a span mean
Span indicates subset of vector space which is of intrest
What happens when you multiply a vector with a scalar value more than one
this expands the vector
What happens when you multiply a vector with a scalar value between 0 and 1
we are compressing the vector
What happens when you multiply a vector with a scalar value less than 1
it changes the direction of the vector
What is the basis vector
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
What is the difference between vector and matrices in relation to multiplication
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
What is linear independence
A set of vector is set to be linear independence if sum of vectors equals to zero,
this means it has a trivial solutioin
What does it mean when the vectors has a trivial solution
This means that the vectior is linear independenta
What does it mean a vector is linear dependent
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
how to find linear dependence using p and n values
if p >n then its linear dependent, else we need to check
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
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
what is vector multiplication of matrices
if the multiplication of a vector result in a vector its called as vector multiplication of matrices
What does Image. Preimage mean inrelation to linear transfomration
preimage - input
image - output
how do we correlate domain and range in relation to linear transformation
range is the all possible images, domain is all possible preimages
What are conditions to call a transfomation linear
vector addition and scalar multiplication
T(u+v) = Tu +Tv
Scalar multiplication
T(cv) = Tc * v
What does Linear transformation do to a vector
Linear Transformation either stretches, compresss, flip or rotate the vector
in Linear transfomation of a vector what does not change
the origin of the vector doesnot change
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
Relation between Linear tranformation and matrix
Linear transformation is nothing but vector multiplication in respect to matrices
Which transformation is called as Algebric and which is geomentric
Linear trans is algerbric, vector trans is geometric
What is onto in linear transformation
if every range equates to every domain then its onto
What is one onto in linear transformation
if every one range equates to every one domain then its one onto
What is not onto
if every range equates to only one domain then its not onto
