Rank of a Matrix: Maximum number of linearly independent row or column vectors.(see pinned comment)

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Row Rank is defined as: Maximum number of Linearly independent row vectors.
Column Rank is defined as: Maximum number of Linearly independent column vectors.
Theorem: Row rank of a matrix is always equal to Column rank of a matrix.
Thus,
Rank of a matrix is defined as row rank = column rank of a matrix.

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20 май 2021

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Комментарии : 55

@DrMathaholic 3 года назад

Row Rank is defined as: Maximum number of Linearly independent row vectors. Column Rank is defined as: Maximum number of Linearly independent column vectors.

@punamgupta7180 2 года назад

for the last question: rank is three and 1st,3rd and 4th row will be linearly independent. thanku sir

@DrMathaholic 2 года назад

Great.. Thank you for posting your answer 😊

@JosephHelbing Год назад

Great explanation thank you

@DrMathaholic Год назад

Thank you and welcome 🙏 🙂

@shahriar6472 2 года назад

Thanks Bro It was helpful.

@DrMathaholic 2 года назад

Welcome:)

@nighatlone77 3 года назад

Consider the system of linear equations AX=B ,then which of the following is true. 1)AX=B is homogenous if B=0 2)if B=0 then detA is not equat to 0 and and x=0 is the solution of AX=B 3)the equation AX=0 has a non trivial solution iff detA=0 4) All of these. Please which is ryt option 1 or 4

@DrMathaholic 3 года назад

1st option is correct. 2nd option is incorrect, detA may or may not b 0.

@DrMathaholic 3 года назад

If A is given to be square matrix then 3rd option is also correct

@nighatlone77 3 года назад

Thanks

@DrMathaholic 3 года назад

@@nighatlone77 welcome!!

@sachinrath219 2 года назад

pl explain bit more clearly on how to know if two rows are linearly indepndent,is it like if we multiply a row by a number we get the other row.

@DrMathaholic 2 года назад

Yes..correct

@sachinrath219 2 года назад

@@DrMathaholic does that mean if the first row is (2,3,5) and the second (4,6,10),then they are dependent i.e 2(2,3,5)= ((4,6,10) or it has some extra logic.

@DrMathaholic 2 года назад

@@sachinrath219 yes..what you wrote is correct!! If a vector can be obtained from other vectors by some linear combination then those vectors are linearly dependent.

@sachinrath219 2 года назад

@@DrMathaholic hope some 'linear combination' means all the elements of a row are to be multiplied by a particular number only like I took 2 as an example.

@sachinrath219 2 года назад

pl intimate how to know which ones should be zero in a 5 x4 matrix for gauss elimination method,i have seen videos where they just tell these should be zero but did not explain why those positions should be zero,thanks.

@DrMathaholic 2 года назад

Nice question. It requires a detailed answer. I will try to type over weekend

@sachinrath219 2 года назад

@@DrMathaholic can u share ur whatsapp no so that i can give a que on gauss elimination what m unable to do,thanks.

@DrMathaholic 2 года назад

@@sachinrath219 pls email me jatinmajithia@gmail.com

@sachinrath219 2 года назад

@@DrMathaholic have mailed the question,pl go thru,thanks.

@sachinrath219 2 года назад

sir,have mailed a question,pl go thru,thanks.

@aashishmeena8183 8 месяцев назад

Air please give answer of these questions 1. No of linearly independent vector 2. Dimensions of soln space 3. Dimensions of null space 4.No. of free variable 5 no of linear independent soln 6 no of lineary independent variable I'm facing difficulties in finding difference blw these..

@DrMathaholic 8 месяцев назад

For which matrix you are having these doubts?

@charusheeladeshpande3717 3 года назад

You take any non zero row. It is always lin ind. Does it mean that rank is 1 for all nonzero matrices???

@irshadsirslectures4446 3 года назад

That need not b linearly independent

@charusheeladeshpande3717 3 года назад

@@irshadsirslectures4446 what need not be lin ind? All I insist is understand the difference betn no. of and max. no. of. They are different.

@irshadsirslectures4446 3 года назад

@@charusheeladeshpande3717 u mean if we take a mtrx , then a non zero row is L. I

@deeplaxmisingh1242 Год назад

Awesome session

@DrMathaholic Год назад

Thanks 😊

@deeplaxmisingh1242 Год назад

Hello sir please tell What is the relation between rank of matrix and independent vectors in vector space

@deeplaxmisingh1242 Год назад

Reply sir

@DrMathaholic Год назад

If you have n vectors and you want to check whether they are independent then simply write them column wise,you get a matrix and now find the rank of this matrix if it is n then vectors r independent else dependent

@nighatlone77 3 года назад

if A is a 5*7 matrix with all its entries equal to -1. The rank of the matrix.....please answer it

@DrMathaholic 3 года назад

As all entries are -1 so all rows and all columns are same. That means we have only one independent row/column and all other rows / columns are 1 times the that row/column. Thus rank is 1.

@nighatlone77 3 года назад

If A is 3*4 real matrix and AX=B is an inconsistent system of linear equations. Then the highest possible rank of A is

@DrMathaholic 3 года назад

@@nighatlone77 no solution means r(A) not equal to r(A|B) , if r(a)=3 then obviously r(a|b) is also 3 and we will have a solution. IF r(A)=2 then we may have r(a|b) =3 and hence inconsistent.

@DrMathaholic 3 года назад

So max possible rank of A can be 2

@nighatlone77 3 года назад

Thank u very much

@mahidhoni8724 3 года назад

Answer : rank of Matrix =2 Linearly independent row vectors = (1 2 3 4) (7 3 4 2) Linearly independent column vectors=(1 3 7 0) (2 6 3 0)

@DrMathaholic 3 года назад

Thank you Dhoni's fan for posting the answer 😀

@mahidhoni8724 3 года назад

@@DrMathaholic 😂👍

@tinyasira6132 2 года назад

how to find the maximum number?

@DrMathaholic 2 года назад

If you have matrix of order 10x10 and suppose that, when you do row operations you get 4 non-zero rows. Then question is what is the maximum number of non zero rows? Answer is 4. Thus 4 will be the rank of that matrix.

@akshitaprajapat225 2 года назад

Sir actually I didn't get that ,,,,how to find column rank , Sir please help me 🙏

@akshitaprajapat225 2 года назад

Could you please explain it to me

@DrMathaholic 2 года назад

Hi Try to see again from 9:00 to 10:00 time frame. After doing row operations simple see how many pivot entries are there.. that is your column rank that means see how many leading non zero entries you have

@DrMathaholic 2 года назад

Or if it's difficult then simply find row rank bcoz row rank is always equal to column rank