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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