Question 1

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

Accepted Answer

Yes, a solution exists.

Question 2

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

Accepted Answer

The answer of Express 
   as a linear...

Question 3

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

Accepted Answer

A) True 
 B)False  True

Question 4

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

Accepted Answer

Linearly i

Question 5

If 
   
, 
   
, and 
   are vectors, then 
   
.

Accepted Answer

A) True 
 B)False

Question 6

If 
   for every 
   
, then the columns of 
   are linearly independent.

Accepted Answer

A) True 
 B)False

Question 7

Suppose there exists a vector 
   such that 
   
. Then the columns of 
   are linearly independent.

Accepted Answer

A) True 
 B)False

Question 8

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

Accepted Answer

Linearly i

Question 9

Determine if the columns of the given matrix span R2.

Accepted Answer

Yes, the c

Question 10

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

Accepted Answer

The answer of The general solution to a linear system...

Question 11

Find the unknowns in the given vector equation.

Accepted Answer

The answer of Find the unknowns in the given vector...

Question 12

Suppose a matrix 
   has 
   rows and 
   columns, with 
   
. Then the columns of 
   do not span Rn.

Accepted Answer

A) True 
 B)False

Question 13

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

Accepted Answer

The answer of Express 
   as a linear...

Question 14

If 
   and 
   are vectors, and 
   and 
   are scalars, then 
   
.

Accepted Answer

A) True 
 B)False

Question 15

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

Accepted Answer

For exampl

Question 16

If 
   
, then 
   
.

Accepted Answer

A) True 
 B)False

Question 17

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

Accepted Answer

The answer of Express the given system of linear equations...

Question 18

Suppose a matrix 
   has 
   rows and 
   columns, with 
   
. Then the columns of 
   span Rn.

Accepted Answer

A) True 
 B)False

Question 19

Determine if the homogeneous system 
   has any nontrivial solutions, where 
   
.

Accepted Answer

has nontri

Question 20

Suppose a matrix 
   has 
   rows and 
   columns, with 
   
. Then the columns of 
   are linearly independent.

Accepted Answer

A) True 
 B)False

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

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

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

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

If , , and are vectors, then .

If for every , then the columns of are linearly independent.

Suppose there exists a vector such that . Then the columns of are linearly independent.

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

Determine if the columns of the given matrix span R².

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

Find the unknowns in the given vector equation.

Suppose a matrix has rows and columns, with . Then the columns of do not span Rⁿ.

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

If and are vectors, and and are scalars, then .

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

If , then .

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

Suppose a matrix has rows and columns, with . Then the columns of span Rⁿ.

Determine if the homogeneous system has any nontrivial solutions, where .

Suppose a matrix has rows and columns, with . Then the columns of are linearly independent.

Systems of Linear Equations

Matrices

Subspaces

Determinants

Eigenvalues and Eigenvectors

Vector Spaces

Orthogonality

Linear Transformations

Inner Product Spaces

Additional Topics and Applications

Filters

Exam 2: Euclidean Space

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

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

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

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

If , , and are vectors, then .

If for every , then the columns of are linearly independent.

Suppose there exists a vector such that . Then the columns of are linearly independent.

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

Determine if the columns of the given matrix span R2.

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

Find the unknowns in the given vector equation.

Suppose a matrix has rows and columns, with . Then the columns of do not span Rn.

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

If and are vectors, and and are scalars, then .

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

If , then .

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

Suppose a matrix has rows and columns, with . Then the columns of span Rn.

Determine if the homogeneous system has any nontrivial solutions, where .

Suppose a matrix has rows and columns, with . Then the columns of are linearly independent.

Systems of Linear Equations

Matrices

Subspaces

Determinants

Eigenvalues and Eigenvectors

Vector Spaces

Orthogonality

Linear Transformations

Inner Product Spaces

Additional Topics and Applications

Filters

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

Determine if the columns of the given matrix span R².

Suppose a matrix has rows and columns, with . Then the columns of do not span Rⁿ.

Suppose a matrix has rows and columns, with . Then the columns of span Rⁿ.