Deck 2: Euclidean Space

Question 1

Determine

, where

Answer 1

Question 2

Express the given vector equation as a system of linear equations.

Answer 2

Question 3

Express the given vector equation as a system of linear equations.

Answer 3

Question 4

Express the given system of linear equations as a single vector equation.

Question 5

Express the given system of linear equations as a single vector equation.

Question 6

The general solution to a linear system is given. Express this solution as a linear combination of vectors.

Question 7

The general solution to a linear system is given. Express this solution as a linear combination of vectors.

Question 8

Find the unknowns in the given vector equation.

Question 9

Find the unknowns in the given vector equation.

Question 10

Express
as a linear combination of the other vectors, if possible.

Question 11

Express
as a linear combination of the other vectors, if possible.

Question 12

If
and
are vectors, and
and
are scalars, then

.

Question 13

If

,

, and
are vectors, then

.

Question 14

If

, then

.

Question 15

Sketch the graph of
and

, and then use the Parallelogram Rule to sketch the graph of

.

Question 16

Determine how to divide a total mass of 18 kg among the vectors
so that the center of mass is

Question 17

Find an example of a linear system with two equations and three variables that has
as the general solution.

Question 18

Find four vectors that are in the span of the given vectors.

Question 19

Find five vectors that are in the span of the given vectors.

Question 20

Determine if
is in the span of the other given vectors. If so, write
as a linear combination of the other vectors.

Question 21

Determine if
is in the span of the other given vectors. If so, write
as a linear combination of the other vectors.

Question 22

Find

,

, and
such that
corresponds to the given linear system.

Question 23

Find

,

, and
such that
corresponds to the given linear system.

Question 24

Express the given system of linear equations as a vector equation.

Question 25

Determine if the columns of the given matrix span R².

Question 26

Determine if the columns of the given matrix span R³.

Question 27

Determine if the system
(where
and
have the appropriate number of components) has a solution for all choices of

.

Question 28

Find all values of
such that the vectors span R².

Question 29

For what value(s) of h do the given vectors span

?

Question 30

Suppose a matrix
has
rows and
columns, with

. Then the columns of
do not span Rⁿ.

Question 31

Suppose a matrix
has
rows and
columns, with

. Then the columns of
span Rⁿ.

Question 32

If the columns of a matrix
with
rows and
columns do not span Rⁿ, then there exists a vector
in Rⁿ such that
does not have a solution.

Question 33

If the columns of a matrix
with
rows and
columns spans Rⁿ, then

.

Question 34

Determine if the given vectors are linearly independent.

Question 35

Determine if the given vectors are linearly independent.

Question 36

Determine if the given vectors are linearly independent.

Question 37

Determine if the columns of the given matrix are linearly independent.

Question 38

Determine if the columns of the given matrix are linearly independent.

Question 39

Determine if the columns of the given matrix are linearly independent.

Question 40

Determine if the homogeneous system
has any nontrivial solutions, where

Question 41

Determine if the homogeneous system
has any nontrivial solutions, where

.

Question 42

Determine by inspection (that is, with only minimal calculations) if the given vectors form a linearly dependent or linearly independent set. Justify your answer.

Question 43

Determine if one of the given vectors is in the span of the other vectors.

Question 44

Suppose matrix
has
rows and
columns, with

. Then the columns of
are linearly dependent.

Question 45

Suppose a matrix
has
rows and
columns, with

. Then the columns of
are linearly independent.

Question 46

Suppose there exists a vector
such that

. Then the columns of
are linearly independent.

Question 47

If
for every

, then the columns of
are linearly independent.

Question 48

If

,

, and
are all linearly independent, then
is linearly independent.