Linear Algebra Chapter 4 T/F Quiz

Null Space and Column Space

• The null space of matrix A is defined as the set of solutions for the equation Ax = 0.
• The null space of an m x n matrix is in R^n, reflecting the number of columns in the matrix.
• The column space of A represents the range of the mapping x -> Ax, consisting of all possible outputs of the transformation.

Consistency and Column Space

• If the equation Ax = b is consistent, it does not guarantee that the column space of A is R^m; this depends on whether every b in R^m has a solution.
• Col A is the set of all vectors expressible as Ax for some x, encompassing all vectors that can be produced through linear combinations of A's columns.

Kernels and Linear Transformations

• The kernel of a linear transformation is a vector space, specifically a subspace of V.
• Nul A refers to the kernel of the mapping x -> Ax, which is defined by the equation Ax = 0.

Range of Linear Transformations

• The range of a linear transformation from V to W is a subspace of W, thus classifying it as a vector space.

Homogeneous Linear Equations

• The solutions of a homogeneous linear differential equation correspond to the kernel of the related linear transformation.

Linear Dependence and Bases

• A single vector is only considered linearly dependent if it is the zero vector; otherwise, it is independent.
• If H is defined as Span{b1,...., bp}, the set {b1,..., bp} may not necessarily be a basis for H.
• The columns of an invertible n x n matrix always form a basis for R^n, supporting the Invertible Matrix Theorem.

Spanning Sets and Basis Size

• A basis is not merely a spanning set; it is a linearly independent set that cannot be enlarged without losing its independence.
• Elementary row operations do not alter the linear dependence relationships among the columns of a matrix.

Subspace Bases

• A linearly independent set within a subspace H does not automatically qualify as a basis unless it spans H.
• If a finite set S of nonzero vectors spans a vector space V, a subset of S is guaranteed to be a basis for V.

B-coordinates

• Assuming B is a basis for a vector space V, any vector x in V represented relative to B yields a coordinate vector in R^n if B contains n vectors.