Let V be the subspace of defined by the equation
Find a linear transformation T from to such that and im(T) = V. Describe T by its matrix A.
The basis of the subspace V of defined by the equation is and the matrix of the linear transformation T from to is
Let V be a subspace such that
Since the basis of the subspace V= , thus the dimension of the subspace V is three.
Also, the vectors in the basis of V are linearly independent so, we have
Thus if T is a linear transformation fromto such that then
The basis of the subspace V of defined by the equationis localid="1659870465922" and the matrix of the linear transformation T from to is
Let A and B be two matrices of the same size, with , both in reduced row-echelon form. Show that . Hint: Focus on the first column in which the two matrices differ, say, the kth columns and of A and B, respectively. Explain why at least one of the columns and fails to contain a leading 1. Thus, reversing the roles of matrices A and B if necessary, we can assume that does not contain a leading 1. We can write as a linear combination of preceding columns and use this representation to construct a vector in the kernel of A. Show that this vector fails to be in the kernel of B. Use Exercises 86 and 87 as a guide.
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