Question

Which of the following are subspaces of \(\mathbf{M}_{22}\) ? Support your answer. a. \(U=\left\\{\left[\begin{array}{ll}a & b \\ 0 & c\end{array}\right] \mid a, b,\right.\) and \(c\) in \(\left.\mathbb{R}\right\\}\) b. \(U=\left\\{\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \mid a+b=c+d ; a, b, c, d\right.\) in \(\left.\mathbb{R}\right\\}\) c. \(U=\left\\{A \mid A \in \mathbf{M}_{22}, A=A^{T}\right\\}\) d. \(U=\left\\{A \mid A \in \mathbf{M}_{22}, A B=0\right\\}, B\) a fixed \(2 \times 2\) matrix e. \(U=\left\\{A \mid A \in \mathbf{M}_{22}, A^{2}=A\right\\}\) f. \(U=\left\\{A \mid A \in \mathbf{M}_{22}, A\right.\) is not invertible \(\\}\) g. \(U=\left\\{A \mid A \in \mathbf{M}_{22}, B A C=C A B\right\\}, B\) and \(C\) fixed \(2 \times 2\) matrices

Short answer

Options (a) and (c) are subspaces; others are not.

Step-by-step solution

Understand Subspace Criteria

A subset of a vector space is a subspace if it satisfies three properties: it is non-empty, it is closed under addition, and it is closed under scalar multiplication.

Analyze Option (a)

The set includes matrices of the form \( \begin{bmatrix} a & b \ 0 & c \end{bmatrix} \). It is non-empty. Adding two such matrices and multiplying by scalar still results in matrices of the same form, so it satisfies all subspace criteria.

Analyze Option (b)

This set includes matrices where \( a+b = c+d \). The set is non-empty. However, adding two matrices from this set, \( (a+b) + (c+d) \) may not satisfy \( a'+b'=c'+d' \), thus it is not closed under addition. This is not a subspace.

Analyze Option (c)

This set includes symmetric matrices \( A = A^T \). It is non-empty. Sum of two symmetric matrices is symmetric, and a scalar multiple of a symmetric matrix is symmetric. It is a subspace.

Analyze Option (d)

The set includes matrices \( A \) such that \( A \cdot B = 0 \). While closed under addition, zero matrix is there, a scalar multiple of such a matrix might not satisfy condition since \( \alpha A \cdot B \) may not equal zero. This is not a subspace.

Analyze Option (e)

The set includes matrices that are idempotent \( A^2 = A \). Zero satisfies \( 0^2 = 0 \). However, adding two idempotent matrices or scalar applying does not always result in another idempotent matrix, failing subspace criteria.

Analyze Option (f)

The set includes non-invertible matrices. Zero matrix is non-invertible so the set is non-empty. Adding two non-invertible matrices can result in invertible one; fails closure in addition.

Analyze Option (g)

Given matrices \( B \) and \( C \); condition \( BA \cdot C = CA \cdot B \) involves specific matrix products. Remains a subset by constraints, not naturally closed under addition or scalar multiplication; fail subspace test.

Key concepts

Vector Space

A vector space is a mathematical structure formed by a collection of vectors, equipped with two operations: vector addition and scalar multiplication. Vectors in a vector space can be added together or scaled by numbers (scalars) from a particular field, usually the field of real numbers, to produce another vector in the same space.

Key properties of a vector space include:

• Closure under addition: Adding any two vectors from the space results in another vector in the same space.
• Closure under scalar multiplication: Multiplying any vector by a scalar yields another vector in the same space.
• Existence of a zero vector: There is a vector (often denoted as 0) that serves as an additive identity.
• Existence of additive inverses: For any vector in the space, there is another vector that, when added together, result in the zero vector.

Understanding these properties is crucial when investigating whether a given set is a subspace of a vector space.

Matrix Operations

Matrix operations are fundamental in linear algebra and involve various manipulations on matrices such as addition, scalar multiplication, and multiplication between matrices. These operations are used extensively when working with vector spaces and subspaces, particularly when matrices represent linear transformations.

Here are important matrix operations:

• Matrix Addition: Matrices of the same dimension can be added element-wise.
• Scalar Multiplication: A matrix can be multiplied by a scalar, which involves multiplying each element of the matrix by the scalar.
• Matrix Multiplication: For matrices, the number of columns in the first matrix must match the number of rows in the second. The product is a new matrix formed by the sum of products over the shared dimensions.
• Transpose: A matrix transpose involves swapping rows with columns, resulting in a new matrix.

These operations are used to assess closure conditions and other subspace criteria.

Linear Transformation

A linear transformation is a mapping between two vector spaces that preserves the operations of addition and scalar multiplication. In the context of matrices, a linear transformation can be represented as a matrix action on a vector, transforming one vector in a space to another in potentially the same or different space.

Important characteristics of linear transformations include:

• Linearity: The transformation respects addition, i.e., \( T(u + v) = T(u) + T(v) \), and scalar multiplication, i.e., \( T(cu) = cT(u) \).
• Representation by Matrices: If the transformation is between finite-dimensional spaces, it can be represented by a matrix.

Understanding linear transformations can help in analyzing if specific matrices form a subspace, particularly when ensuring operations still result in a valid transformation within the space.

Subspace Criteria

To determine if a subset of a vector space is a subspace, it must meet specific criteria. Examining these conditions can help identify which sets of matrices from a given exercise form valid subspaces. The criteria for a subspace are:

• Non-emptiness: The subset must contain the zero vector, ensuring it is not empty.
• Closure under Addition: The sum of any two vectors from the subset must also belong to the subset.
• Closure under Scalar Multiplication: Multiplying a subset vector with any scalar should result in another vector that is still in the subset.

By checking each of these criteria, one can determine if given matrix conditions satisfy being a subspace of a larger space, like \( extbf{M}_{22} \), the vector space of all \(2 \times 2\) matrices.