MAS 2103 Midterm: MAS 2103 FAU Exam 2S07

TRUE-FALSE, Determine whether each statement below is either true of false. Write either TRUE or FALSE (all caps), as appropriate, before each one. a) ____ A linear system with a reduced row echelon form having a row of zeroes has no solutions. b) ____ No set of 6 dependent vectors in R^5 spans R^5. c) ____ Let V be a vector space and S be a subset of V. If V = span(S), then s is a basis for V. d) _____ If u middot v = 0 then either u = 0 or v = 0. e) _____ If v is a 5-component vector of norm 4, then ||3v|| = (3)^5 4 f) _____ The solution set of a consistent nonhomogeneous linear sysem Ax = b, where A is an m times n matrix and b is m times 1, is a subspace of R^n g) ____ A linear system with a reduced row echelon form having a row of zeroes has infinitely many solutions. h) _____ If u, v and w are all 5-component vectors then (u middot v) middot w = u middot (v middot w) h) ____ If u, v and w are all 5-component vectors then (u middot v) middot w = u middot (v middot w) i) ____ The general solution of the nonhomogeneous linear system Ax = b can be obtained by adding b to the general solution of the homogeneous linear system Ax = 0.

True/False: Explain why the statement is true, or give a specific counter-example: The span of {0-} is {0-}. It v- , w- are both vectors in R2 that are non-zero and are not parallel to each other then Span[ { v-, w-} ] is all of R2. If v-, w- are both vectors in R2 that arc non-zero and one is a scalar multiple of the other then Span[ {v-, w-} ] is the same as Span[ { v- }] . If v- belongs to the span of set of vectors S, then so does for every scalar c. Any set of vectors S containing the zero vector must be linearly independent. The Zero vector is in the span of any non-empty set of vectors S If a subset of R4 contains more than 4 vectors and is linearly dependent There is a subset of P4 that contains more than 4 vectors and is linearly independent If Span[ ] , then there is a nontrivial solution to the equation

For what values of h will the vectors in each of the following sets he linearly independent? Briefly explain your answer in each case. [1 -2 1],[2 1 -1],[-3 6 h] [1 2],[-1 2],[0 h] [0 1 1 0],[1 0 1 -1],[h -1 0 1] For each of the following statements, indicate whether it is always true, sometimes true, or never true: The columns of a matrix A are linearly independent if Ax = 0 has the trivial solution. If v1, v2 and v3 are in R3 and v3 is not a linear combination of v1 and v2, then the set {v1, v2, v3} is linearly independent. If {v1 .. v } is a set of linearly independent vectors in R then n m. For each of the following, give an example or explain why it is impossible to construct one: A 2 times 3 matrix whose columns span R3 A 2 times 3 matrix whose columns span R2 A 3 times 3 matrix whose columns span R3 A 3 times 2 matrix whose columns span R3 A 3 times 2 matrix whose columns span R2 Let A = [1 2 -1 -1 1 -5 2 -1 8] and observe that three times the first column minus two tunes the second column equals the third column. Without reducing the matrix, find a non-trivial solution to the homogeneous equation Ax = 0. Give a geometric description of the following transformations x rightarrow Ax: A = [2 0 0 1] A = [-1 2 0 1] A = [ /2 /2 /2 /2]