True or false? 6. If B is a row-echelon form of a matrix A, then the nonzero columns of B form a basis for the column space of A.

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rakesh05's profile pic

rakesh05 | High School Teacher | (Level 1) Assistant Educator

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False.

Because B is a row reduced matrix not the column reduced. If you want to find the basis for the column space, first of all reduce B to a column-echelon form   C (say). Then the non zero columns of C will form a basis for the column space.

e.g.   Consider the row-echelon matrix `B=[[1,2,0,3,4],[0,1,2,4,0],[0,0,0,0,0],[0,0,0,0,0]]` .

Now we see that there are five non-zero columns. But they are not linearly independent and hence cannot form a basis at this stage.

Now we convert this matrix B to a column reduced echelon matrix C.

   `[[1,0,0,0],[2,1,0,0],[0,2,0,0],[3,4,0,0],[4,0,0,0]]` `~~` `[[1,0,0,0],[0,-1,0,0],[0,2,0,0],[0,-4,0,0],[0,0,0,0]]` `~~` `[[1,0,0,0],[0,1,0,0],[0,2,0,0],[0,4,0,0],[0,0,0,0]]` `~~[[1,0,0,0],[0,1,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]`

So our matrix C=`[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,0],[0,0,0,0,0]]` .

We see that C has only two non zero columns. So these two non zero columns will form the basis for the column space of the matrix A as mentioned in the given problem.

pramodpandey's profile pic

pramodpandey | College Teacher | (Level 3) Valedictorian

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False.

The column spaces of A and U are generally unequal .

Let us consider matrix A=`[[1,2],[2,4]]`

Row echelon form of A,say B=`[[1,2],[0,0]]`

rank (B)=rank(A)=1

In this example, here two non zero  columns, namely`([1],[0])`  and `([2],[0])` are linearly dependent.So will not basis for column space.

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