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]

Question

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]

Laaibah Pitt · Accepted Answer

Step 1  Given:  A=[131240]  The reduced row echelon form of A=[131240] is [10−2011]  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,  x1−2x3=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 x∈N(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)

Jeffrey Jordon · Accepted Answer

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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}

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