Is Set H A Subspace Of M2x2? Justification And Explanation

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In the realm of linear algebra, vector spaces and their subspaces form the bedrock upon which many concepts and applications are built. Understanding the properties and characteristics of subspaces is crucial for grasping the structure and behavior of linear transformations, systems of linear equations, and other fundamental topics. In this comprehensive exploration, we delve into a specific question concerning a set of matrices and its potential to form a subspace within the larger vector space of 2x2 matrices. Specifically, we will investigate whether the set H, comprising all matrices of the form

[ 2a  b ]
[ 3a+b 3b ]

where a and b are real numbers, qualifies as a subspace of M2x2, the vector space of all 2x2 matrices with real entries. To embark on this investigation, we will first lay the groundwork by revisiting the definition of a subspace and the conditions that a subset must satisfy to be considered a subspace. Then, we will meticulously examine the set H, scrutinizing its elements and their interactions under the operations of matrix addition and scalar multiplication. By carefully verifying whether H adheres to the subspace criteria, we will arrive at a definitive conclusion regarding its subspace status. This exploration will not only provide an answer to the specific question at hand but also serve as a valuable exercise in applying the fundamental principles of linear algebra and solidifying our understanding of subspaces.

Keywords: subspace, matrices, vector space, linear algebra, matrix addition, scalar multiplication, M2x2, linear combinations, zero matrix

Subspace Criteria

Before we dive into the specifics of the set H, it's essential to have a clear understanding of what constitutes a subspace. A subspace is a subset of a vector space that itself satisfies the axioms of a vector space. To determine if a subset is a subspace, we need to verify three crucial conditions:

  1. The Zero Vector: The zero vector of the parent vector space must be present in the subset. This ensures that the subset has a neutral element for addition.
  2. Closure Under Addition: If two vectors are members of the subset, their sum must also be a member of the subset. This guarantees that the addition operation within the subset remains consistent.
  3. Closure Under Scalar Multiplication: If a vector is a member of the subset, the product of any scalar and that vector must also be in the subset. This ensures that scaling vectors within the subset does not lead to vectors outside the subset.

If a subset satisfies all three of these conditions, it is deemed a subspace of the original vector space. Failure to meet even one condition disqualifies the subset from being a subspace. These conditions are not arbitrary; they stem directly from the vector space axioms and ensure that a subspace inherits the essential algebraic structure of its parent vector space. By verifying these criteria, we can rigorously determine whether a given subset possesses the necessary properties to function as a self-contained vector space within a larger vector space.

Analyzing the Set H

Now, let's turn our attention to the set H, which is defined as the set of all 2x2 matrices of the form:

[ 2a  b ]
[ 3a+b 3b ]

where 'a' and 'b' are real numbers. Our mission is to determine whether this set H is a subspace of M2x2, the vector space of all 2x2 matrices with real entries. To accomplish this, we must systematically verify the three subspace criteria outlined earlier: the presence of the zero vector, closure under addition, and closure under scalar multiplication. Each criterion will be examined in detail, with careful attention paid to the structure of the matrices in H and how they behave under the respective operations.

  1. Zero Vector: To check for the zero vector, we need to see if there exist real numbers 'a' and 'b' such that the resulting matrix is the 2x2 zero matrix:
[ 0 0 ]
[ 0 0 ]

By setting a = 0 and b = 0, we obtain the zero matrix:

[ 2(0)  0 ] = [ 0 0 ]
[ 3(0)+0 3(0) ]   [ 0 0 ]

Therefore, the zero matrix is indeed an element of H.

  1. Closure Under Addition: Let's consider two arbitrary matrices in H:
A = [ 2a1     b1    ]
    [ 3a1+b1   3b1   ]

B = [ 2a2     b2    ]
    [ 3a2+b2   3b2   ]

where a1, b1, a2, and b2 are real numbers. Their sum is:

A + B = [ 2a1+2a2      b1+b2        ]
        [ 3a1+b1+3a2+b2  3b1+3b2       ]

      = [ 2(a1+a2)       (b1+b2)         ]
        [ 3(a1+a2)+(b1+b2) 3(b1+b2)    ]

Notice that the resulting matrix is also in the form required for membership in H, with the real numbers (a1 + a2) and (b1 + b2) playing the roles of 'a' and 'b', respectively. Thus, H is closed under addition.

  1. Closure Under Scalar Multiplication: Let's take an arbitrary matrix from H:
A = [ 2a    b   ]
    [ 3a+b  3b  ]

and multiply it by a scalar 'c', where 'c' is a real number:

cA = [ c(2a)    cb    ]
     [ c(3a+b)  c(3b) ]

   = [ 2(ca)     cb      ]
     [ 3(ca)+cb 3(cb)   ]

Again, the resulting matrix maintains the required form for membership in H, with (ca) and (cb) as the real number parameters. This demonstrates that H is closed under scalar multiplication.

Conclusion

Having meticulously examined the set H against the three subspace criteria, we can now confidently draw a conclusion. We have established that the zero vector is a member of H, and that H exhibits closure under both matrix addition and scalar multiplication. Since H satisfies all three conditions, it qualifies as a subspace of M2x2. This means that H itself forms a vector space, inheriting the vector space structure from the larger space M2x2. The significance of this finding lies in the fact that we can now apply the full power of linear algebra tools and techniques to analyze the set H, understanding its properties and behavior within the broader context of 2x2 matrices. This exploration underscores the importance of the subspace concept in linear algebra, allowing us to identify and work with subsets that possess the essential characteristics of vector spaces.

Keywords: conclusion, subspace criteria, zero vector, closure under addition, closure under scalar multiplication, M2x2, vector space, linear algebra

Is the Set of Matrices a Subspace? A Justification

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