Linear Algebra Exam 1 Concept Problems

If a system of linear equations has more variables than equations, then the system must have infinitely many solutions : False. The system can be inconsistent. For example: x + y + z = 5, x + y + z = 2.

If A is an m × n matrix and A has a pivot in every column, then the equation Ax = b has a solution for each b in R m. : False. For example, if A = (1,0,0,1,0,0) , then Ax = (0,0,1) is inconsistent

If the reduced row echelon form of an augmented matrix has a row of zeros, then the system of linear equations corresponding to the augmented matrix has infinitely many solutions. : False. It could be inconsistent, or it could have a unique solution.

If A is an m×n matrix and Ax = b has a unique solution for some b in R m, then Ax = 0 has only the trivial solution. : True. Since Ax = b is consistent, its solution set is a translation of the solution set to Ax = 0. Since Ax = b has a unique solution, this means Ax = 0 has a unique solution (namely the trivial solution).

If A is a 3 × 4 matrix and b is a vector so that the set of solutions to Ax = b is a line through the origin, then b =0. : True. If the solution set contains the origin then x = 0 is a solution so b = A(0) = 0. Very similar to a 3.3-3.4 supplemental problem

If A is a 3 × 3 matrix and Ax = 0 has infinitely many solutions, then Ax = b must be inconsistent for some b in R 3 . : True. If Ax = 0 has infinitely many solutions then A has at most two pivots. Since A has three rows, this means A cannot have a pivot in every row, so Ax = b must be inconsistent for some b in R 3

If A is an m × n matrix and m > n, then then there is at least one vector b in R m which is not in the span of the columns of A. : True. The matrix A has at most n pivots, but it has m rows and m > n so it cannot have a pivot in every row

Are there three nonzero vectors v1 , v2 , v3 in R 3 so that Span{v1 , v2 , v3 } is a plane but v3 is not in Span{v1 , v2 }? If your answer is yes, write such vectors v1 , v2 , v3 and label each vector clearly. : For example, v1 = (1 0 0), v2 = (−1 0 0) , v3 = (0 1 0).

Is there a 2 × 2 matrix A so that the solution set for the equation Ax = 0 is the line x1 = x2 + 1? If yes, write such an A. If no, justify why there is no such A : No. The solution set to Ax = 0 must include the origin x1 = x2 = 0, which is not on the line x1 = x2 + 1.

Suppose w1=(1 0 0), w2=(0 1 2), and w3=(-2 3 5). Then span{w1,w2,w3} is R3. : True. Pivot in every row so vectors span R3.

Is there a 2x2 matrix A so that the solution set to Ax=0 is the line x1-x2=3? If your answer is yes, write such a matrix A. If answer is no, justify why there is no such matrix A. : No. x=0=(0 0) must be a solution to Ax=0 but 0-0 does not equal 3.

If b is a vector in R3, then b must be a linear combination of the vectors (1 0 -1), (1 1 0), and (0 1 1). : False. These vectors don't span R3.

What best describes the geometric relationship between the solutions to Ax=0 and the solutions to Ax=b?(a) they are both lines through the origin(b) they are parallel lines(c) they are both planes through the origin(d) they are parallel planes : (b) they are parallel lines

Give an example of a matrix A so the solutions to Ax=0 form a line in R4. : A = (1 0 0 0, 0 1 0 0, 0 0 1 0)

Suppose A is a 2x2 matrix and the set of solutions to Ax=0 is the line y=x. Also suppose that b is a nonzero vector in R2. Which of the following can possibly be the set of solutions to Ax=b? Select all that apply.(a) the line y=x(b) the line y=x+1(c) the line x=0(d) the origin : (b) the line y=x+1

Is the augmented matrix (0 1 1 1, 0 0 0 1) in reduced row echelon form? : No.

Suppose A is a 2x2 matrix and A(1 1) = (19 7). Is it possible that the set of solutions to Ax=0 is the line x1=x2? : No.

Suppose A is a 4x5 matrix. Is it possible that Ax-b is consistent for all b in R4? : Yes.

There is a 5x7 matrix A so that Ax=b is consistent for every b in R5. : True. A can be a 5x7 matrix with a pivot in every row.

Suppose we have three equations in three variables. Which of the following can be the set of solutions? Select all that apply.(a) a point(b) a plane(c) a line(d) no solution : Can be all options.