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Math 341 self quiz

 

Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not
true in every case.


a. Every matrix is row equivalent to a unique matrix in echelon form.

b. Any system of n linear equations in n variables has at most n solutions.

c. If a system of linear equations has two different solutions, it must have infinitely many solutions.

d. If a system of linear equations has no free variables, then it has a unique solution.

e. If an augmented matrix [A b ] is transformed into [C d ] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets.

f. If a system Ax = b has more than one solution, then so does the system Ax = 0.

g. If A is an m×n matrix and the equation Ax = b is consistent for some b, then the columns of A span Rm.

h. If an augmented matrix [A b ] can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent.

i. If matrices A and B are row equivalent, they have the same reduced echelon form.

j. The equation Ax = 0 has the trivial solution if and only if there are no free variables.

k. If A is an m×n matrix and the equation Ax = b is consistent for every b in Rm, then A has m pivot columns.

l. If an m×n matrix A has a pivot position in every row, then the equation Ax has a unique solution for each b in Rm.

m. If an n×n matrixAhas n pivot positions, then the reduced echelon form of A is the n×n identity matrix.

n. If 3×3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations.

o. If A is an m×n matrix, if the equation Ax = b has at least two different solutions, and if the equation Ax = c is consistent, then the equation Ax = c has many solutions.

p. If A and B are row equivalent m×n matrices and if the columns of A span Rm, then so do the columns of B.

q. If none of the vectors in the set S = {v1, v2, v3} in R3 is a multiple of one of the other vectors, then S is linearly independent.

r. If {u, v,w} is linearly independent, then u, v, and w are not in R2.

s. In some cases, it is possible for four vectors to span R5.

t. If u and v are in Rm, then −u is in Span{u, v}.

u. If u, v, and w are nonzero vectors in R2, then w is a linear combination of u and v.

v. If w is a linear combination of u and v in Rn, then u is a linear combination of v and w.

w. Suppose that v1, v2, and v3 are in R5, v2 is not a multiple of v1, and v3 is not a linear combination of v1 and v2. Then {v1, v2, v3} is linearly independent.

x. A linear transformation is a function.

y. If A is a 6×5 matrix, the linear transformation x → Ax cannot map R5 onto R6.

z. If A is an m×n matrix with m pivot columns, then the linear transformation x → Ax is a one-to-one mapping.

2. Let a an b represent real numbers. Describe the possible solution sets of the (linear) equation ax =b. [Hint: The number of solutions depends upon a and b

3. The solutions (x, y, z )of a single linear equation

ax + by + cz = d


form a plane in R3 when a, b, and c are not all zero. Construct sets of three linear equations whose graphs (a) intersect in a single line, (b) intersect in a single point, and (c) have no
points in common . Typical graphs are illustrated in the figure.

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