number 3 Given the rref([1 2 3 2 4 6 3 6 9]) = [1 0 0 2 0 0 3 0 0]). State a bas

Question

number 3

Given the rref([1 2 3 2 4 6 3 6 9]) = [1 0 0 2 0 0 3 0 0]). State a basis for the vector space v-Span[1 2 3],[2 4 6]= the vectors used to construct the space, v Circle a single 61 from basis set, and explain your answer. State the matrix, A, representing a linear transformation T:R^3 rightarrow R^3, that maps the vectors in R^3 to their (mirror images) with respect to z-axis. Show your work. Given the multiplicative inverse of A = [1 2 3 1 0 1 0 0 1]), A^-1 = [0 1 -1 1/2 -1/2 -1 0 0 1], Consider a linear transformation T defined by the matrix A. State the dimension of the range of T, Explain your answer and circle a single answer.

Explanation / Answer

3. Since TX = Ax, the range of T is same as Col(A). To determine Col(A), we will reduce A to its RREF as under:

Add -2 times the 1st row to the 2nd row

Add -3 times the 1st row to the 3rd row

Multiply the 2nd row by -1/2

Add 2 times the 2nd row to the 3rd row

Add -1 times the 2nd row to the 1st row

Then the RREF of A is I3. This implies that the columns of A are linearly independent. Hence a basis for Range(T)= Col(A) is { (1,2,3)T,(1,0,1)T,(0,0,1)T}.

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