Linear Algebra Quiz # 1 Review / Fall 06

 

(1.) Determine if "b" is a linear combination of the vectors formed by the columns of the matrix "A".

 

       

 

Form the augmented matrix,, and reduce it echelon form.

 

     

 

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(2.) Let , , and .

For what values of "h" is "y" in the plane generated by "" and ""?

 

Form the augmented matrix, , and reduce it echelon form.

 

           

 

The vector "y" is in the plane formed by the two vectors or in  provided that the system is consistent,

which it is for  or .

 

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(3.) Let  and . Show that the equation  does not have a solution for all

possible "b" and describe the set of all "b" for which  does have a solution.

 

Form the augmented matrix, , and reduce it echelon form.

 

            

 

The equation  is consistent only if .

The set of all such "b" is a plane thru the origin in .

 

 

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(4.) Let  and . Is "u" in a subset of  spanned by the columns of "A"?

 Why or why not?

 

Form the augmented matrix, , and reduce it echelon form.

 

           

 

The equation  has no solution, so "u" is not spanned by the columns of "A".

 

 

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(5.) Let  Do columns of "B" span ?

Does  have a solution for each "y" in ?

 

If  has only the trivial solution, , then columns of "B" span  because they are linearly independent. This is equivalent to "B" having all columns as pivot columns. We determine that now.

 

        

 

Since not every column is a pivot column, "B" does not span  and therefore not every "y" can be constructed from a linear combination of the columns of "B". Accordingly, the equation  does NOT have a solution for every "y" in .

 

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(6.) Describe all solutions of the following system in parametric vector form. Also, give a geometric

description of the solution set.

 

 

 

 

Form the augmented matrix, , and reduce it echelon form.

 

 

                

 

 

 

The solution set is a line in 3-space passing thru the point:  and parallel to the line that is the solution set of the homogeneous equation.

 

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(7.) Find a parametric equation of the line "M" thru  and .

 

 

The two parametric equations are equivalent.

 

 

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(8.) Balance this chemical reaction:

----->

 

            

 

         

 

 

 

 

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Let . Then here is the balanced chemical reaction:

----->

 

(9.) Find values of "h" for which the vectors are linearly dependent. Justify your answer.

 

              

Do there exist non-zero weights, "", such that: ? If so, then the vectors are Linearly Dependent.

 

The vectors are linearly dependent if there is more than the trivial solution to the matrix equation . Row reduce the augmented matrix,

 

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Since  is a free variable, there is more than the trivial solution for all values of "h" and thus the vectors are Linearly Dependent for all values of "h".

 

(10.) How many rows and columns must a matrix "A" have in order to define a mapping from  into  by the rule ?

The vector "x" has 4 rows and its image  must have 5 rows. Therefore, the matrix "A" must have 5 rows and 4 columns.

 

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(11.) With "T" defined by , find a vector "x" whose image under "T" is "b", and determine whether "x" is unique.

 

     

 

Solve the matrix equation  and choose any solution "x".

 

 

         

 

 

 

Choose any one vector from that infinite set, say . Obviously, "x" is not unique.

 

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(12.) Assume that "T" is a Linear Transformation. Find the standard matrix "A" of "T". T: ---> rotates points about the origin thru  radians clockwise.

 

     

 

 

 

 

 

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(13.) Assume that "T" is a Linear Transformation. Find the standard matrix "A" of "T". T: ---> is a vertical shear transformation that maps "" into "" but leaves the vector "" unchanged. Here "a" and "b" are scalars.

 

         

 

 

    

 

 

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(14.) Determine if the specified linear transformation is (a.) one-to-one and (b.) onto. Justify your answer. . Note that "" are not vectors but are entries in vectors.

 

   

 

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