Algebra: Rank, Row, and Column Space

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Questions and Answers

What does the column space of a matrix A represent?

The intersection of the row and column spaces

The subspace spanned by the column vectors of A (correct)

The set of all row vectors of A

The transpose of the row space of A

If matrix A is transformed into matrix B using elementary column operations, which of the following statements is true?

The column space of A will change

The rank of A will be different from the rank of B

The row space of A will remain the same (correct)

The row space of A will become the column space of B

Which of the following properties is true about the row space and column space of a matrix?

Row space is a subset of the column space

They have the same dimension (correct)

Row space and column space can have different dimensions

Row space is always larger than column space

What is the rank of a matrix A?

The dimension of the column space or row space of A Signup and view all the answers

Which of the following best describes a basis for the row space of a row-echelon matrix R?

The nonzero rows of R Signup and view all the answers

For the matrix A =

            [1 3]
            [2 2]
            [3 0], what is the dimension of col(A)?

2 Signup and view all the answers

Which of the following statements about matrix transformations and their effects is false?

Transformations can increase the rank of a matrix Signup and view all the answers

What can be concluded about the row vectors of matrix A given that A is m × n?

They span a space in Rn Signup and view all the answers

What is true regarding the rank of matrix A?

rank(A) ≤ min{m, n} Signup and view all the answers

Which property of a null matrix is accurate?

Its rank is zero Signup and view all the answers

How is the nullity of matrix A related to its rank?

dim(null(A)) = n - rank(A) Signup and view all the answers

Which statement is correct about the image space of matrix A?

dim(im(A)) = r where r is the rank of A Signup and view all the answers

When do the rows of matrix A span Rn?

When rank(A) = n Signup and view all the answers

Which property is false regarding rank and invertibility?

If rank(A) = m, then A is always invertible Signup and view all the answers

What is the implication of Ax = 0 having only the trivial solution?

The columns of A are linearly independent Signup and view all the answers

What does the statement rank(A + B) ≤ rank(A) + rank(B) imply?

It reflects a general limitation on rank addition Signup and view all the answers

Study Notes

Algebra Notes

The subject is algebra, taught by Associate Professor Nguyen Trung Thanh at the Faculty of Data Science and Artificial Intelligence.

Rank of a Matrix

A matrix's rank is the dimension of its row space (or column space).
It's denoted as rank(A).
The rank of a matrix is equal to the dimension of its row space and its column space.

Column Space

The column space (col(A)) of a matrix A is the set of all possible linear combinations of its columns.
It's a subspace of ℝ^m.
Example: If A has column vectors x₁ and x₂, then col(A) = Span{x₁, x₂}.

Row Space

The row space (row(A)) of a matrix A is the set of all possible linear combinations of its rows.
It's a subspace of ℝⁿ.
Example: For a matrix A with rows y₁, y₂, y₃, the row space is row(A) = Span{y₁, y₂, y₃}.

Matrix Properties

If matrix A transforms into B using elementary row operations, then row(A) = row(B).
If matrix A transforms into B using elementary column operations, then col(A) = col(B).
Nonzero rows of a row-echelon matrix R are a basis of row(R).
Columns of R with leading ones are a basis of col(R).
The row space and column space of an m × n matrix A have the same dimension (rank(A)).

Properties of Rank

The rank of a matrix A is equal to the rank of its transpose A^T.
The rank of a zero matrix is zero.
The rank of an identity matrix of order n is n.
The rank of matrix A is less than or equal to the minimum of m and n (where m is the number of rows and n is the number of columns).
If U and V are invertible matrices, then rank(AU) = rank(VA) = rank(A).
If A transforms into B by elementary row/column operations, then rank(A) = rank(B).
The rank of the sum of two matrices A and B is less than or equal to the sum of their ranks.
The rank of the product of two matrices AC is less than or equal to the minimum of rank(A) and rank(C).

Null Space

The null space of a matrix A, denoted as null(A), is the set of all vectors x that satisfy Ax = 0.
The dimension of the null space is called the nullity of A.

Image Space

The image space of a matrix A, denoted by im(A), is the set of all possible outputs (Ax) when the matrix operates on all possible input vectors x.

Other Properties

Rank(A) = n, if and only if the rows of A span ℝⁿ and the columns of A are linearly independent in ℝ^m.
If the matrix A is an m × n matrix, and CA = I for an n × m matrix C, then the matrix A is invertible.
If Ax = 0, and x is in Rⁿ, then x = 0.
If m = n (square matrix), then det(A) ≠ 0 if and only if rank(A) = n.
The system Ax = b is consistent for every b in ℝ^m.

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Description

This quiz covers key concepts in algebra, focusing on the rank of a matrix, its row space, and column space. Designed for students in the Faculty of Data Science and Artificial Intelligence, this material is crucial for understanding matrix properties and their applications. Test your knowledge and reinforce your grasp of these foundational concepts.

Algebra: Rank, Row, and Column Space

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Questions and Answers

What does the column space of a matrix A represent?

If matrix A is transformed into matrix B using elementary column operations, which of the following statements is true?