MTH207 Lesson 4: Vector Spaces and Bases

Flashcards covering Vector Spaces and subspaces, Null Spaces, Column Spaces, Linear Transformations, Linearly Independent Sets, Bases, Dimension of a Vector Space, and Rank based on MTH207 Lesson 4 notes.

A set of vectors {V1, V2,…,Vp} in a vector space is said to be linearly independent if the vector equation C1V1 + C2V2 + … + CpVp = 0 has only the __ solution c1 = 0,…, Cp = 0.

trivial

A set of vectors is linearly dependent if there exist weights c1,…,Cp, not all 0, such that C1V1 + C2V2 + … + CpVp = __.

0

A set containing the __ is linearly dependent.

zero vector

A set of two vectors is linearly dependent if and only if one is a __ of the other.

multiple

A set with only one non-zero vector is __.

linearly independent

A basis set is an "efficient" spanning set containing no __ vectors.

unnecessary

An indexed set of vectors "beta" = {b1,…, bp} in V is a basis for a subspace H if it is a __ set and H = Span{b1,…,bp}.

linearly independent

The Invertible Matrix Theorem states that for a square n x n matrix A, the statements regarding A are either all true or all __.

false

According to the IMT, if A is an invertible matrix, then the columns of A form a __ set.

linearly independent

According to the IMT, if A has n pivot positions, then A is __.

an invertible matrix

To determine if a set of vectors is a basis for R^3, one can form a matrix A with these vectors as columns and check if it has __ pivots using row reduction.

3

To find a basis for Nul A, one must __ the augmented matrix [A 0] and express the general solution in parametric vector form.

row reduce

A basis can be constructed from a spanning set of vectors by __ vectors which are linear combinations of preceding vectors.

discarding

To find a basis for Col A, identify the pivot columns in the row-reduced echelon form, and then use the corresponding __ of the original matrix A.

pivot columns

Elementary row operations on a matrix do not affect the linear __ relations among the columns of the matrix.

dependence