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HARVARD MATH 21B - First midterm

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Math 21b: First MidtermPlease show your work for all problems except for the True/False questions on thelast page. If you need additional space, feel free to use the backs of the pages (pleasemake a note if you do so, so we know to look). Each of the four questions will be worthten points. No calculators, notes, books, or any other aids are allowed.Please don’t write on this front page (except for your name) as we will use it torecord grades.Name:#1#2#3#4Total11. For an arbitrary constant k, consider the linear systemx − 2kz = 0x + 2y + 6z = 22z − kx = 1of equations in x, y, z.(a) [4 pts.] For k = 2 the system has a unique solution. Find it, and check that it is actuallya solution.(b) [4 pts.] For which value(s) of k does the system not have a unique solution?(c) [2 pts.] For the value(s) of k for which the system does not have a unique solution, whichif any yield a system with no solution, and which if any yield a system with infinitelymany solutions?22. (a) [4 pts.] Compute the row-reduced echelon form (rref) of the matrix0 0 1 1 0 00 2 1 0 1 03 2 1 0 0 1.(b) [4 pts.] Your computation in (a) gives the inverse of a 3 × 3 matrix A. What is A, andwhat is A−1?(c) [2 pts.] Use the matrices A and A−1of (b) to solve the linear system represented byA~x =~b, where~b =3−12.Check your work by verifying that the entries of ~x actually satisfy that linear system.33. Let ~v1, ~v2, ~v3be the following vectors in R4:~v1=1452, ~v2=1673, ~v3=−1211.Let V be the linear subspace of R4spanned by ~v1, ~v2, ~v3.(a) [2 pts.] Write a matrix whose image is V.(b) [5 pts.] Find a basis of V.(c) [3 pts.] Is the vector−1432in V ?44. For each of the following 10 assertions, circle T if the assertion is true, and circle F if theassertion is false. Each is worth one point. For this question only, there is no need to justifyyour answers.T F a) If the linear system A~x =~b has a unique solution then A must be a square matrix.T F b) Reflection about the line x + y = 1 is a linear transformation of R2.T F c) A linear transformation from Rnto Rmalways has infinitely many vectors in thekernel if n > m.T F d) If A is a matrix such that AAAAA = I2then A is invertible.T F e) If A is an invertible matrix and B is a matrix such that AB is a zero matrix, thenB is a zero matrix.T F f) If A and B are invertible n × n matrices then (A + B)(A − B) = A2− B2.T F g) Suppose a linear system has coefficient matrix A and augmented matrix B. If thesystem is consistent then rank(A) = rank(B).T F h) If the vector ~u is a linear combination of vectors ~v and ~w, then ~w must be a linearcombination of vectors ~u and ~v.T F i) If vectors ~v1, ~v2, ~v3, ~v4are linearly independent, then vectors ~v2, ~v3, ~v4must belinearly independent as well.T F j) The setxy: x is an integer(infinitely many vertical lines, see picture below)is closed under vector


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HARVARD MATH 21B - First midterm

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