Linear Algebra Basis & Dimension Flashcards

Questions and Answers

What is a basis of a subspace?

A set of vectors that is both spanning and linearly independent (correct)

Any set of vectors in the subspace

A single vector in the subspace

None of the above

What does the dimension of a subspace represent?

The number of vectors in any basis of the subspace.

The pivot columns of a matrix A form a basis for the column space of A.

True

The free variables in the solution set of Ax = 0 do not contribute to the basis of the null space Nul(A).

False Signup and view all the answers

What is the Rank Theorem?

The rank of a matrix is the dimension of its column space and is related to nullity. Signup and view all the answers

What does the Basis Theorem state?

Any m linearly independent vectors in a subspace of dimension m form a basis for that subspace. Signup and view all the answers

How can a matrix be viewed in the context of functions?

A matrix can be seen as a function that maps input vectors to output vectors. Signup and view all the answers

What is an identity transformation?

It is the transformation defined by IdRⁿ(x) = x. Signup and view all the answers

What does the matrix transformation associated with a matrix A do?

It transforms a vector x in Rⁿ to the vector Ax in Rm. Signup and view all the answers

The range of a matrix transformation is always the entire space Rm.

False Signup and view all the answers

Study Notes

Basis of a Subspace

A basis of a subspace V in Rⁿ consists of vectors {v₁, v₂,..., vm} that both span V and are linearly independent.

Dimension

The dimension of a subspace V, denoted as dimV, is determined by the number of vectors in any basis of V.

A Basis for the Column Space

The pivot columns of a matrix A serve as a basis for its column space, Col(A).

A Basis for the Null Space

The vectors associated with free variables in the solution of Ax = 0 form a basis for the null space, Nul(A).

The Rank Theorem

The rank of matrix A, noted as rank(A), is the dimension of Col(A).
The nullity of matrix A, nullity(A), is the dimension of Nul(A).
Relationships: rank(A) = dim Col(A) (number of pivot columns) and nullity(A) = dim Nul(A) (number of free variables).
For an m x n matrix A: rank(A) + nullity(A) = n.

The Basis Theorem

For a subspace V of dimension m:
- Any set of m linearly independent vectors in V is a basis for V.
- Any set of m vectors that spans V is also a basis for V.

Matrices as Functions

A matrix A with m rows and n columns can be viewed as a function mapping input vector x (in Rⁿ) to output vector b (in Rm) through the equation b = Ax.
The set of all possible output vectors from this transformation forms the range dictated by the solutions of Ax = b.

Transformation

A transformation (or map) T assigns each vector x in Rⁿ to a vector T(x) in Rm.
Rⁿ refers to the domain of the transformation while Rm is the codomain.
The collection of all images from T forms the range of T.

Identity Transformation

The identity transformation IdRⁿ maps every vector x in Rⁿ to itself, defined as IdRⁿ(x) = x, ensuring no alteration occurs to the input vector.

Matrix Transformation

For an m x n matrix A, the associated matrix transformation T: Rⁿ → Rm is defined by T(x) = Ax, translating input vector x in Rⁿ to the output vector Ax in Rm.

Matrix Transformation Continuation

The domain of transformation T corresponds to Rⁿ (where n equals the number of columns of A).
The codomain relates to Rm (where m is the number of rows of A).
The range of transformation T corresponds specifically to the column space of A.

Description

Explore the concepts of Basis and Dimension, along with the Rank and Basis Theorems in this quiz. Test your understanding of linear transformations and how they relate to subspaces in vector spaces. Perfect for students studying linear algebra.