Additional Exercises for Chapter 3: Big Ideas about Vectors
Review Exercises
Here are some vectors.
Describe the span of these vectors as the span of as few vectors as possible.
Here are some vectors.
Describe the span of these vectors as the span of as few vectors as possible.
Here are some vectors.
Describe the span of these vectors as the span of as few vectors as possible.
Here are some vectors.
Now here is another vector:
Is this vector in the span of the first four vectors? If it is, exhibit a linear
combination of the first four vectors which equals this vector, using as few vectors as
possible in the linear combination.
Here are some vectors.
Now here is another vector:
Is this vector in the span of the first four vectors? If it is, exhibit a linear
combination of the first four vectors which equals this vector, using as few vectors as
possible in the linear combination.
Here are some vectors.
Now here is another vector:
Is this vector in the span of the first four vectors? If it is, exhibit a linear
combination of the first four vectors which equals this vector, using as few vectors as
possible in the linear combination.
Are the following vectors linearly independent? If they are, explain why and if they
are not, exhibit one of them as a linear combination of the others. Also give a linearly
independent set of vectors which has the same span as the given vectors.
Are the following vectors linearly independent? If they are, explain why and if they
are not, exhibit one of them as a linear combination of the others. Also give a linearly
independent set of vectors which has the same span as the given vectors.
Are the following vectors linearly independent? If they are, explain why and if they
are not, exhibit one of them as a linear combination of the others. Also give a linearly
independent set of vectors which has the same span as the given vectors.
Are the following vectors linearly independent? If they are, explain why and if they
are not, exhibit one of them as a linear combination of the others. Also give a linearly
independent set of vectors which has the same span as the given vectors.
Are the following vectors linearly independent? If they are, explain why and if they
are not, exhibit one of them as a linear combination of the others. Also give a linearly
independent set of vectors which has the same span as the given vectors.
Are the following vectors linearly independent? If they are, explain why and if they
are not, exhibit one of them as a linear combination of the others. Also give a linearly
independent set of vectors which has the same span as the given vectors.
Here are some vectors in .
These vectors can’t possibly be linearly independent. Tell why. Next obtain a
linearly independent subset of these vectors which has the same span as these
vectors.
Here are some vectors in .
These vectors can’t possibly be linearly independent. Tell why. Next obtain a
linearly independent subset of these vectors which has the same span as these
vectors.
Here are some vectors in .
These vectors can’t possibly be linearly independent. Tell why. Next obtain a
linearly independent subset of these vectors which has the same span as these
vectors.
Challenge Exercises
Here are some vectors in .
These vectors can’t possibly be linearly independent. Tell why. Next obtain a
linearly independent subset of these vectors which has the same span as these
vectors.
Suppose , , and are the rows of the matrix . Then we can interpret the solution to
the system of equations as the intersection of three planes containing the origin.
Discuss what this intersection would look like geometrically if the reduced row
echelon form of is of the form:
- (a)
- (b)
- (c)
- Are there any other possibilities?
Octave Exercises
Use Octave to check your work on Problems prb:3.8 to prb:3.10. The first steps of prb:3.8 are in the code.
See if you can interpret the result to answer the question.
To use Octave, go to the Sage Math Cell Webpage, copy the code below into the cell, select OCTAVE as the language, and press EVALUATE.
% Is b in the span of the other vectors?
v1=transpose([1 0 -2]);
%transpose turns this into a column matrix
v1
v2=transpose([1 1 -2]);
v3=transpose([2 -2 -3]);
v4=transpose([-1 4 2]);
b=transpose([-1 -4 -2]);
M=[v1 v2 v3 v4 b]
R=rref(M)
% After viewing R, I added the following to check my work:
19*v1-12*v2-4*v3
Use Octave to check your work on Problems prb:3.15 to prb:3.27. The first steps of prb:3.15 are
done in the Octave window. See if you can interpret the result to answer the
question.
To use Octave, go to the Sage Math Cell Webpage, copy the code below into the cell, select OCTAVE as the language, and press EVALUATE.
% Test for linear independence
v1=transpose([1 3 -1 1]);
v2=transpose([1 4 -1 1]);
v3=transpose([1 4 0 1]);
v4=transpose([1 10 2 1]);
A=[v1 v2 v3 v4]
R=rref(A)
% After viewing R, I added the following to check my work:
-6*v1+4*v2+3*v3
Bibliography
The Review Exercises come from the end of Chapter 4 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)
Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, pp. 151–152, 220–237.
The Challenge Exercises come from the end of Chapter 1 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)
W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 33–34.