Linear Algebra: Null Space and Column Space

Questions and Answers

How can you find if a vector belongs to N(A)?

Check if Ax gives you the zero vector.

How do you check if a vector belongs to R(A)?

Check if the vector gives you a consistent REF when augmented with the vectors as columns.

What is the process to find a basis for N(A)?

Use free variables in REF([A | O]).

How can you find a basis for R(A)?

Use pivot columns in REF(A) and relate them to the columns of the original matrix. Signup and view all the answers

What is the method for finding the row space basis?

Take the REF and select the rows with pivots. Signup and view all the answers

How do you check if a set of vectors spans R^n?

Find REF and check if the REF has n pivot columns. Signup and view all the answers

What steps are involved in finding a basis for a set of vectors?

1.) Create a matrix with vectors as columns, 2.) Use pivot columns in REF(A), 3.) Relate to vectors of the original set. Signup and view all the answers

What is the method to check if a vector belongs to the span of a set of vectors?

1.) Augment the vector to the original set, 2.) See if REF is consistent. Signup and view all the answers

How can you find the basis for a subset of S that is the basis for Sp(S)?

1.) Stack the set to form a matrix, 2.) Find REF, 3.) Identify leading 1/pivot columns. Signup and view all the answers

Study Notes

Null Space (N(A))

A vector belongs to N(A) if multiplication by matrix A results in the zero vector.
To verify, check if Ax = 0.

Range/Column Space (R(A))

To determine if a vector is in R(A), augment it with the matrix columns and verify whether the resulting system is consistent in REF.

Basis for Null Space (N(A))

Find a basis by examining the free variables in the reduced echelon form of the augmented matrix [A | O].
The steps include finding REF for the matrix, expressing the solutions as vectors, and selecting vectors corresponding to the free variables.

Basis for Column Space (R(A))

Utilize pivot columns identified in REF(A) to form a basis.
Columns from the original matrix corresponding to the pivot columns establish the basis for the column space.

Basis for Row Space

The basis for the row space is derived from the rows in REF that contain pivot elements.
Select only those rows that feature a leading entry (pivot).

Spanning R^n

To check if a set of vectors spans R^n, obtain REF and confirm the presence of n pivot columns.
The existence of n pivot columns indicates a complete spanning.

Basis of a Set of Vectors

Create a column matrix from the given vectors.
Identify pivot columns in REF(A) to determine the basis, which directly corresponds to original vectors in the set.

Vector in the Span

To check if a vector lies within the span of given vectors, augment it to the original set and determine if the resulting REF is consistent.
A consistent REF implies the vector is indeed spanned by the set.

Basis for a Subset of Vectors

Form a matrix using the set of vectors and find REF.
Columns corresponding to leading 1s indicate the basis elements from the original set.
Identify non-basis columns by relating rows in REF to original columns to establish which columns are extra.

Description

This quiz covers concepts related to null space, range (column space), and row space in linear algebra. It focuses on how to determine bases for these spaces and their properties through matrix operations. Test your understanding of finding bases using reduced echelon form and consistent systems.