For each of the following matrices, determine a basis for each of the subspaces R(AT ), N(A), R(A), and N(AT ): A=begin{bmatrix}1 & 3&1 2 & 4&0end{bmatrix}

Albarellak 2020-12-01 Answered
For each of the following matrices, determine a basis for each of the subspaces R(AT ), N(A), R(A), and N(AT ):
A=[131240]
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Expert Answer

Laaibah Pitt
Answered 2020-12-02 Author has 98 answers
Step 1
Given:
A=[131240]
The reduced row echelon form of A=[131240] is [102011]
Since (-1,0,-2) and (0,1,1) form a basis for the row space of matrix A , we have {(1,0,2)T,(0,1,1)T} form a basis for R(AT)
When from the reduced row echelon form of matrix A , we have,
x12x3=0
x1=2x3
x1+x3=0
x1=x3
Step 2
Set x3=α. Then N(A) consists of all vectors of the form α(2,1,1)T
Therefore, (2,1,1)T is its basis
Now AT=[123410]
The reduced row echelon form of AT=[123410] is [100100]
Since (1,0) and (0,1) form the basis for the row space of matrix AT , we have {(1,0)T,(0,1)T} form a basis for R(A).
When xN(AT) from the reduced row echelon form of matrix AT , we have,
x1=0 and x2=0
Step 3 It follows that N(AT)=0
Therefore , there is no basis for N(AT)
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Jeffrey Jordon
Answered 2022-01-30 Author has 2047 answers

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