Linear Algebra Problems for Unit 1 TRUE or FALSE Expect roughly five questions like these to appear on the exam. You will be asked to justify all your answers! 1. If A is a matrix in reduced echelon form, then at least one entry in every column of A must be 1. 2. The matrix is in reduced echelon form. 3. No set of two vectors in R3 can span R3. 4. If two m × n matrices are row equivalent, they have the same number of pivot positions. 5. If two m × n matrices have the same pivot positions, then they must be row equivalent. 6. There exist three vectors in R3, no two scalar multiples of each other, whose span is a line through the origin. 7. There exist three vectors in R3, no two scalar multiples of each other, whose span is a plane through the origin. 8. If A is an m × n matrix whose colums span Rm, then the equation A~x = ~b is consistent for every . 9. If S and T are both linear transformations from R2 to R2, then S ◦ T is also a linear transformation from R2 to R2. 10. If S and T are functions from R2 to R2 that are both nonlinear, then the composition S ◦ T must also be nonlinear. 11. There is a linear transformation T : R → R that sends the interval [−1,1] to the interval [1,3]. 12. There is a set of 2015 vectors in R2011 that is linearly independent. 13. The transformation is linear. 1 14. There is a linear transformation T : R2 → R1 that is one-to-one but not onto. ALWAYS, SOMETIMES, or NEVER Expect roughly five questions like these to appear on the exam. You will be asked to justify all your answers! 1. A linear system of n equations in m unknowns has an augmented matrix of size m×(n+1). 2. If ~b is a vector in Span{a~1,...,a~p}, then the vector equation ~b = x1a~1 + ... + xna~n has a unique solution. 3. A linear transformation T : Rn → Rm can be represented as a matrix transformation. 4. The function A(x) := ax + b is a linear transformation from R1 to R1. 5. Suppose and we know that . Then we can solve the equation . 6. Suppose T : R4 → R4 is a one-to-one linear transformation and A is its standard matrix. Then every row in A contains a pivot. 7. If a linear transformation T : Rn → Rm sends some nonzero vector (in Rn) to the zero vector (in Rm), then its standard matrix contains a column of all zeros. 8. The span of a linearly dependent set of vectors {~a,~b,~c} in R3 contains two linearly independent vectors. 9. An m×n matrix A sends a linearly independent set of vectors to a linearly dependent set of vectors, but the equation A~x = ~0 has only one solution (namely ~x = ~0). 10. A linearly dependent set of vectors {~v1,~v2} in R7 can be made into a linearly independent set {~v1,~v2,~v3} by adding another (carefully chosen) vector, ~v3 ∈ R7. Short Answer Expect roughly five questions like these to appear on the exam. Be sure to read each question carefully. 1. Suppose we are studying the flow of traffic along a number of one-way streets, as illustratedin the diagram below, where each arrow gives the direction of travel and each label gives the flow 2 along that section of street in vehicles per hour. Find the general solution for the unknown flows x1, x2, x3, and x4, and give any necessary conditions on any free variables that appear to guarantee that all the flows will be positive. 100 ◦ x1 • x3 ◦ x2 ◦ x4 200 2. Find all values of h and k such that the vectors (a) span R3; (b) are linearly independent. 1 3 4 t 3. Suppose A is the matrix 4 2 1 3 and ~b is the vector 3 . 8 s 2 (a) For which values of s,t will the system A~x =~b be inconsistent? (b) For which values of s,t will there be exactly one solution to A~x =~b? 4. Suppose A is a matrix for which it is known that and Find a solution to the equation 5. Determine by inspection whether or not the given sets of vectors are linearly independent. Justify each answer. 3 6. Suppose A is an m×n matrix. Explain why A has a pivot in every column if and only if the equation A~x = ~0 has only the trivial solution. 7. Suppose S : R5 → R2 and T : R2 → R5 are linear transformations. The linear transformation (T ◦ S) : R5 → R5 is defined by ( . (a) Explain how you know that S is not one-to-one. (b) Explain how you know that (T ◦ S) is not one-to-one. (c) Suppose A is the standard matrix of the linear transformation (T ◦S). How many rows and columns are there in A? (d) Can every column of A contain a pivot? (e) Can every row of A contain a pivot? (f) Explain how you know that the columns of A don’t span R5. 8. For given a,b,c ∈ R, suppose we define a function fa,b,c(t) := a · et + b · cos(t) + c · sin(t). (a) Find formulas (in terms of a,b,c) for p,q,r ∈ R so that the derivative fa,b,c0 (t) = p · et + q · cos(t) + r · sin(t) (b) Show, using the formulas for p,q,r you found above, that the transformation D given by is linear. (c) Find the standard matrix of D. 4 9. (a) Suppose T : Rn → R3 is a linear transformation and you know and . Determine T(3~v − 2w~). (b) Explain how you know there there cannot be any linear transformation T : R2 → R2 such that , and 10. Find all values (if any) of s and t so that, for the matrix, (a) the matrix transformation determined by A is onto. (b) the matrix transformation determined by A is one-to-one. (c) there exactly one free variable in the matrix equation A~x = ~0. 5 .
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