Linear Algebra Exam 2

What is the transpose of a matrix A?

The matrix Aᵀ obtained by interchanging rows and columns of A.

What is a subspace?

A subset V of ℝⁿ with the following properties:
Contains the zero vector (0 ∈ V)

Closed under scalar multiplication (c ∈ ℝ, v ∈ V ⇒ cv ∈ V)

Closed under addition (u,v ∈ V ⇒ u+v ∈ V)

What are the properties of matrix transposing?

(cA)^T = cA^T
(AB)^T = B^TA^T

What is C(A)?

The column space of A is the span of all the columns of A. It represents all outputs b you could get from Ax = b. If b is in C(A), then Ax = b has a solution.

What is R(A)?

The row space of A is the span of the rows of A.

What is N(A)?

The null space of A is all vectors x that A maps to zero.

What is N(A^T)?

The null space of A transpose is all vectors z that A^T maps to zero.

What is the Span of a set of vectors?

The span of a set of vectors is the set of all possible linear combinations of those vectors.

What are the four special subspaces?

N(A), C(A), R(A), and N(A^T)

What are the relationships of the four subspaces?

N(A) = R(A)^⊥
R(A) = N(A)^⊥

N(A^T) = C(A)^⊥
C(A) = N(A^T)^⊥

How do you find C(A)?

The basis for C(A) is the set consisting of the column vectors that include one of the pivot variables. The dimension of C(A) is the same as R(A).

How do you find R(A)?

The basis for R(A) is all of the non-zero row(s) of A after finding its reduced-echelon form.

How do you find N(A)?

The basis for N(A) is found by performing Gaussian elimination on A, then expressing the pivot variable(s) in terms of the free variable(s) and using the vectors formed by this.

How do you find N(A^T)

The basis for N(A^T) is found by performing Gaussian elimination on A^T, then expressing the pivot variable(s) in terms of the free variable(s) and using the vectors formed by this.

What is linear independence?

Linear independence is when the only solution to c₁v₁ + ··· + cₖvₖ = 0 is c₁=···=cₖ=0.

What is linear dependence?

Linear dependence is when there is a non-trivial solution to c₁v₁ + ··· + cₖvₖ = 0.

What is a basis?

A basis is a minimal set of vectors {v1,…,vk} that:

1. Spans the subspace (every vector in the subspace is a linear combination of them).

2. Is linearly independent (no redundant vectors).

What is dimension?

Dimension is the number of vectors in any basis for V.

What is Rank?

Rank is the number of rows in an echelon form of Matrix A.