Let A be an 3 x 4 matrix, and B an 4 x 3 matrix. Prove: If AB Is, then the columns of B are linearly independent. Let A be an 3 x 4 matrix, and B an 4 x 3 matrix. Prove: If AB Is, then the c...

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Let A be an 3 x 4 matrix, and B an 4 x 3 matrix. Prove: If AB Is, then the columns of B are linearly independent.

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Answer 1

Answer #1

$AB = I_3 = [1,0,0;0,1,0;0,0,1]$

The above statement implies the column of matrix AB is linearly independent

Let the matrix B of the form with three columns

$B = [b_1,b_2,b_3]$

Since the columns of AB is linearly independent, so we can write

$AB = A[b_1,b_2,b_3] = [Ab_1,Ab_2,Ab_3]$

Since these are linearly independent, so we can write it in the form that the only solution of the equation is (c1=c2=c3=0)

$c_1Ab_1 + c_2Ab_2 + c_3Ab_3 = 0$

Taking the A common, we can write

$A[c_1b_1 + c_2b_2 + c_3b_3] = 0$

Now, the matrix A cannot be zero matrix, otherwise the product AB won't be equal to I3, this implies

$c_1b_1 + c_2b_2 + c_3b_3 = 0, when \ (c_1=c_2=c_3=0)$

Hence the columns of matrix B are linearly independent.

Note - Post any doubts/queries in comments section.

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