Briefly explain why each of the following statements is True or False. One point

Question

Briefly explain why each of the following statements is True or False. One point if you know whether it is true or false. Two points for a correct explanation. a. Suppose the vectors c_1, v_2 and v_3 R^3 are linearly dependent. Then they span a place in R^3 b. If A is a matrix with m rows and AB = C for some matrices B and C, then the matrix C must have m rows. c. If A is a 6 times 5 matrix, then the linear transformation which maps x rightarrow Ax cannot be an onto transformation. d. If one of the columns of A is c = (3 -2 1), then c is in the span of the columns of A. e. If QA is an m times n matrix with a pivot in each row, then x rightarrow Ax is a one-to-one mappint. 6. Rewrite the equation [2 -3 0 0 1 -2 3 -4 1] x = (2 -1 3) as a linear combination of vectors.

Explanation / Answer

5 a.

False. Only linearly independent vectors can span a vector space. v1, v2, v3 being linearly dependent cannot span R^3.

5b) True

AB=C means Matrix A has the same no of columns as no of rows in matrix B.

Suppose A = mxn, B is nxp then AB will be mxp

So C will have m rows.

5c) Tx= Ax means A-1T = I

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