Exam 2: Euclidean Space

Determine , where

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

Find the unknowns in the given vector equation.

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

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

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

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

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

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.

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

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

Determine if the columns of the given matrix span R³.

Determine if the given vectors are linearly independent.

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

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

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

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

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

Determine if the given vectors are linearly independent.

Find , , and such that corresponds to the given linear system.