Exam 2 Math 307

4.1 Given vectors u, v, w, in ℝ^3 , the linear combination of these vectors is what?

au + bv + cw. a, b, c = constants

4.1 In ℝ^3, when are vectors u and v dependent?

If and only if they are parallel.

4.1 Are vectors u and v linearly dependent/independent in ℝ^3? u = (1, 2, 3), v = (3, 2, 1)

not parallel, therefore independent

4.1 Are vectors u and v linearly dependent/independent in ℝ^3? u = (-4, 2, 16), v = (2, -1, -8)

Since u = -2v -> they are parallel & dependent

4.1 In ℝ^3, when are vectors u, v, and w linearly independent?

Vectors u, v, w are linearly independent if au + bv + cw =/= 0. If not, they are dependent.

4.1 A vector v in ℝ^3 is what? And what are its components?

A vector v in ℝ^3 is an ordered triple of numbers (a, b, c) and numbers (a, b, c) are the components of vector v.

4.1 Calculate a determinant to determine if u, v, w are linearly dependent/independent

*DONT FORGET TO SWITCH SIGNS WHEN FINDING DETERMINANT

4.1 Use the reduced row echelon method to determine if the vectors are independent/dependent.

4.1 Solve 26.

4.1 Solve 31.

4.1 Solve 32.

4.1

4.2 What are the two trivial subspaces for any vector space V? What are they called?

the zero-space {0}, and V. Subspaces other than the zero-space are called proper subspaces.

4.2 Any subspace must be ______ and contain the _____ vector.

nonempty, zero

4.2

4.2 What is a nontrivial linear combination of vectors?

A nontrivial linear combination is a linear combination of vectors where at least one of the coefficients is not zero. It is the opposite of a trivial linear combination, where all coefficients are zero. A set of vectors is called linearly dependent if there exists a nontrivial linear combination of these vectors that equals the zero vector. 

4.2

Find determinant, use RREF to solve for constants and put vectors in columns INSTEAD of rows.

4.2

4.2