Linear Algebra: Vector Spaces Flashcards

Questions and Answers

Define a subspace.

A subset (S) of a larger set of vectors (V) that contains the zero vector, is closed under addition, and is closed under scalar multiplication.

What is the column space?

Denoted col(A), it is the span of the columns of matrix A.

Define closure.

Something is closed under an operation when the result of said operation produces something still in the set.

The span of a set of vectors is always a subspace of Rⁿ.

True Signup and view all the answers

Define the null space.

Written Nul(A), it is the solution set of the homogeneous equation {x: x ∈ Rⁿ and Ax = 0}. Signup and view all the answers

When is vector x in the null space of A?

When Ax = 0. Signup and view all the answers

Why is the null space a subspace?

It contains the 0 vector, any two vectors chosen sum to a vector that produces 0, and scalar multiplication still produces a vector that produces 0. Signup and view all the answers

Why is the column space a subspace?

The column space is the span of the columns of matrix A, which is always a subspace. Signup and view all the answers

Null space is a subspace of _ and Column space is a subspace of _.

Rⁿ; R^m Signup and view all the answers

Finding a vector in the column space means?

Selecting one of the columns of the matrix. Signup and view all the answers

Finding a vector in the null space means?

Solving Ax = 0 and writing the final solution in vector form as a single vector. Signup and view all the answers

Vector v is in the column space of A if?

If Ax = v is consistent. Signup and view all the answers

How is the basis found for col(A)?

The columns in the original matrix that contain pivots in RREF make up each vector in the basis. Signup and view all the answers

How is the basis found for nul(A)?

Row reduce and solve for each variable (in terms of free variables as needed) to obtain the solution in vector form. Signup and view all the answers

What is a vector space?

A vector space V is a set of vectors that are closed under addition and scalar multiplication. Signup and view all the answers

What are the β-coordinates of V?

v = (x)b₁ + (y)b₂ + (z)b₃. Signup and view all the answers

If Null(A) = { 0 }, then?

The columns are linearly independent, as the homogeneous equation only has the trivial solution. Signup and view all the answers

Define the dimension of a vector space.

The number of vectors in the basis set. Signup and view all the answers

What is another name for rank with regards to dimension?

Rank is another name for dim(colA). Nullity is another name for dim(nulA). Signup and view all the answers

What is the shortcut to determining the dimension of both the null space and column space?

dim(nulA) is equal to the number of free variables; dim(colA) is equal to the number of pivots. Signup and view all the answers

What is the Rank-Nullity theorem?

Rank + Nullity = n. Signup and view all the answers

Define the row space.

The span of each row of the matrix. Signup and view all the answers

How is a basis found for the row space?

Put the matrix in echelon form, and the basis contains each non-zero row. Signup and view all the answers

Given the basis vectors and a particular vector X, how is the coordinate vector found?

By solving Av = X through row reduction and determining the weights. Signup and view all the answers

What is the change of basis matrix equation?

It is expressed as Ax = b and A⁻¹b = x. Signup and view all the answers

Study Notes

Subspace

A subspace is a subset S of a vector space V, satisfying three conditions: includes zero vector, closed under addition, and closed under scalar multiplication.

Column Space

Denoted col(A), the column space is the span of the columns of the matrix A.

Closure

An operation is closed if the result remains within the set after applying the operation.

Span and Subspaces

The span of a set of vectors in Rⁿ is always a subspace of Rⁿ.

Null Space

Null space, denoted Nul(A), consists of solutions to the homogeneous equation Ax = 0.

A Vector in Null Space

A vector x is in the null space of A if Ax = 0, meaning it transforms into the zero vector when multiplied by A.

Null Space as Subspace

The null space is a subspace because it contains the zero vector, closed under vector addition, and closed under scalar multiplication.

Column Space as Subspace

The column space is a subspace since it's the span of the columns of matrix A.

Locations of Spaces

Null space is a subspace of Rⁿ (columns/unknowns), while column space is a subspace of R^m (rows/entries).

Finding Vectors in Spaces

A vector in the column space is found by selecting a column of matrix A.
A vector in the null space is determined by solving Ax = 0.

Consistency in Column Space

A vector v is in the column space of A if the equation Ax = v is consistent.

Basis of Column Space

Basis for col(A) includes columns of the original matrix that contain pivots in reduced row echelon form.

Basis of Null Space

To find the basis for nul(A), row reduce the matrix and express each variable in terms of free variables.

Vector Space Definition

A vector space V is a set of vectors closed under addition and scalar multiplication, often represented as Rⁿ.

β-coordinates of V

If v = (x)b₁ + (y)b₂ + (z)b₃, the scalars represent contributions from each basis vector to form vector V.

Null Space Implication

If Null(A) = { 0 }, columns of A are linearly independent, only the trivial solution exists for the homogeneous equation.

Dimension of Vector Space

The dimension of a vector space is defined by the number of vectors in the basis set, denoted dim(V).

Rank and Nullity

Rank refers to dim(col(A)) and nullity refers to dim(nul(A)).

Shortcuts for Dimensions

The dimension of the null space (dim(nul(A)) equals the number of free variables, while the dimension of the column space (dim(col(A))) equals the number of pivots.

Rank-Nullity Theorem

The Rank-Nullity Theorem states that Rank + Nullity = n, where n is the total number of columns in the matrix.

Row Space

Row space is the span of all rows in the matrix.

Basis for Row Space

To identify the basis for the row space, convert the matrix to echelon form and select each non-zero row.

Finding Coordinate Vector

Given basis vectors and an arbitrary vector X, the coordinate vector is found by solving Av = X.

Change of Basis Matrix Equation

The change of basis can be expressed as Ax = b, leading to x = A⁻¹b, where P sub B represents the matrix of basis vectors.

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Test your knowledge of vector spaces in linear algebra with these flashcards. Each card offers definitions and key concepts like subspace, column space, and closure. Perfect for students looking to strengthen their understanding of vector spaces.