Instructions: Solve all of the problems. For full credit, you must show complete
ID: 3115096 • Letter: I
Question
Instructions: Solve all of the problems. For full credit, you must show completecorrect, legible work. Read carefully before you start working. allowed, but phones, computers, and PDAs are not. Each question is worth equal points No books or notes are allowed. Calculators are 2 1 0 i Solve the linear system Ax = b, where A-1 =1-1 1 0|,xslz2l,and b=151. 0 0 3 7 2. Show that the subset W-(a,b,0)1 a, b are real numbers) S R3 is a subspace of R. 3. Consider the subset of elements in R S= {(1,0,0),(0,2,2), (3,2,2)) (a) Is S a linearly independent set? Justify (b) Does S span R3? Justify (c) Use your answers to explain whether S is a basis for R3. 10 3 4 4. Consider the matrix A 0 1 1 (a) Give a basis for the row space of A. (b) Find the rank of A, rank(A). (c) Find the nullity of A, that is, find the dimension of the nullspace N(A). 5 Verify the differential operator D, : B, defined as Dlae+ a'z+aaz?) = + 2aar, is a linear transformation. (Here, P is the vector space of polynomials of degree at most n.) 6. Consider the linear transformation L:R2R2 defined by L(y)- (2r-y,0) (a) Find the kernel of L (you can describe it using set notation, or geometrically, or as the span of a set of vectors) (b) Find the range of L (again, you may describe it using set notation, geometrically, or as the span of a set of vectors). 7. Find the standard matrix for the linear T:R2 R2 defined by 8. Let T:RR2 be a linear whose standard matrix is A 13 Find the matrix A' for T relative to the basis B'-((2,1), (1, 1)) 9 (BONUS) Let T : R3 be the linear transformation defined by T(e) = 1. Tea)-41, and T(e)-3z2. Show that T is an isomorphism of vector spacesExplanation / Answer
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(a)To determine whether S is a linearly independent set, we will reduce A to its RREF as under:
Multiply the 2nd row by ½
Add -2 times the 2nd row to the 3rd row
Then the RREF of A is
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Now, apparently,(3,2,2)= 3(1,0,0)+1(0,2,2) so that S is not a linearly independent set.
(b) Since none of the columns in the RREF of A have 1 in the 3rd row, hence S does not span R3.
(c ) Since S does not span R3, hence S is not a basis for R3.
Please post the remaining questions again
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