MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Transcription

1 est Review-Linear Algebra Name MULIPLE CHOICE Choose the one alternative that best completes the statement or answers the question Solve the system of equations ) 7x x + + 9x (-,, ) (, -, ) (,, ) (,, ) ) Determine whether the system is consistent ) x + x - + x + - Yes No ) Determine whether the matrix is in echelon form, reduced echelon form, or neither -7 ) - ) Echelon form Neither Reduced echelon form ) - - ) Echelon form Reduced echelon form Neither Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated ) Find the reduced echelon form of the given matrix )

2 he augmented matrix is given for a system of equations If the system is consistent, find the general solution Otherwise state that there is no solution - ) ) x - + is free is free x + - is free x + - x is free 7) - 7 7) x - + is free x is free (-, 7) No solution Find the indicated vector ) Let u - -9, v - Find -u + v ) Solve the problem 9) Let a -, a - -, a, and b - Determine whether b can be written as a linear combination of a, a, and a In other words, determine whether weights x,, and exist, such that x a + a + a b Determine the weights x,, and if possible x,, x -, -, x -,, No solution 9) Compute the product or state that it is undefined ) [ ] )

3 Write the system as a vector equation or matrix equation as indicated ) Write the following system as a vector equation involving a linear combination of vectors x - - x + x - x - x x - + ) x + x - + x - x + x - + x Solve the problem ) Let A and b b b b ) Determine if the equation Ax b is consistent for all possible b, b, b If the equation is not consistent for all possible b, b, b, give a description of the set of all b for which the equation is consistent (ie, a condition which must be satisfied by b, b, b) Equation is consistent for all b, b, b satisfying b + b Equation is consistent for all b, b, b satisfying -b + b Equation is consistent for all possible b, b, b Equation is consistent for all b, b, b satisfying 7b + b + b ) Find the general solution of the simple homogeneous ʺsystemʺ below, which consists of a single linear equation Give your answer as a linear combination of vectors Let and be free variables x - + x x x x - x x (with, free) (with, free) (with, free) (with, free) )

4 ) Find the general solution of the homogeneous system below Give your answer as a vector x + - x x x x - - x x - - ) - ) Determine if the columns of the matrix A - are linearly independent No Yes ) For what values of h are the given vectors linearly dependent? -,, - -, - h Vectors are linearly dependent for h - Vectors are linearly independent for all h Vectors are linearly dependent for h - Vectors are linearly dependent for all h ) ) 7) Let : R -> R be a linear transformation that maps u - Use the fact that is linear to find the image of u + v into - and maps v into - 7) ) Let A -7 and u 7 Define a transformation : R -> R by (x) Ax Find (u), the image of u under the transformation )

5 9) Let A and b - Define a transformation : R -> R by (x) Ax If possible, find a vector x whose image under is b Otherwise, state that b is not in the range of the transformation b is not in the range of the transformation - 9) ) he columns of I are e, e, e ) Suppose that is a linear transformation from R into R such that ( e) -, ( e ), and ( e ) x Find a formula for the image of an arbitrary x in R x x x- x + x + - x -x + x x x- x x + - -x + Find the standard matrix of the linear transformation ) : R -> R first performs a vertical shear that maps e into e + e, but leaves the vector e unchanged, then reflects the result through the horizontal x-axis )

6 Determine whether the linear transformation is one -to-one and whether it maps as specified ) Let be the linear transformation whose standard matrix is - A Determine whether the linear transformation is one-to-one and whether it maps R onto R One-to-one; not onto R One-to-one; onto R Not one-to-one; not onto R Not one-to-one; onto R ) Describe geometrically the effect of the transformation ) Let A Define a transformation by (x) Ax Vertical shear Horizontal shear Projection onto -axis Projection onto x-axis )

7 Answer Key estname: REV ES ) D ) A ) B ) B ) C ) C 7) D ) D 9) D ) B ) C ) C ) B ) B ) A ) D 7) C ) B 9) B ) D ) C ) B ) A 7