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- Suppose the augmented matrix of the matrix equation Ax = b is:
1 0 4 7 2 a 0 1 3 0 4 c 0 0 0 3 6 g 0 0 0 − 5 8 k 0 0 0 0 10 p
1A. This augmented matrix is not quite in reduced row echelon form. Find the reduced row echelon form; circle your final matrix.
1B. Use the answer to (1A) to answer these questions: What conditions are there on a, c, g, k and p so that Ax = b has... 1B-i.... NO solutions. (tell me about all of a, c, g, k and p in each of these three parts).
1B-ii.... exactly one solution.
1B-iii.... infinitely many solutions.
1C: Suppose that a, c, g, k and p are in fact chosen so that Ax = b is consistent. Give the particular solution obtained by setting any and all free variables equal to 2; your answer will be in terms of a, c, g, k and p.
- Suppose a linear transformation T : Rs^ → Rt^ is given by T (x) = Ax, where A is this matrix:
2A. What are s and t? s = t = A =
2B. Find T
2C. Is
∈^ T^?^ Explain your answer.
To answer the next questions, the following fact is useful:
(^) is row equivalent to
2D. What (if any) conditions are there on b 1 , b 2 and b 3 so that b =
b 1 b 2 b 3
(^) is in the image of T?
2E. Verify that b =
(^) does in fact satisfy your conditions in (2D); show the work.
2F. Find all x such that T (x) =
; express your answer in parametric (x = p + vh) form.
- Define T : X → R^3 by T
([
a b
])
9 − ab 0 2 a + 3b
, where X is the largest subset of R^2 for which all
the formulas can be computed.
4A. Find T
([
])
4B. Explicitly describe X.
4C. Find a smaller codomain for T than R^3 ; explain your answer.
4D. Give an explicit counter example that shows why T is not a linear transformation. Explain what you’re doing.
4E. Bonus! Sketch the domain of T.
4F. Bonus! Sketch the image of T. (This will be tough).
