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Chapter 1: Problem 17

Prove that a subset \(W\) of a vector space \(V\) is a subspace of \(V\) if and only if \(\mathrm{W} \neq \varnothing\), and, whenever \(a \in F\) and \(x, y \in \mathrm{W}\), then \(a x \in \mathrm{W}\) and \(x+y \in \mathrm{W}\).

Short Answer

Expert verified
In conclusion, we have proven that a subset \(W\) of a vector space \(V\) is a subspace of \(V\) if and only if \(W\) is nonempty and satisfies the following conditions: for any scalar \(a\) and any vectors \(x\), \(y\) in \(W\), \(ax \in W\) and \(x+y \in W\). This was demonstrated by showing that these conditions are equivalent to the three defining properties of a subspace, which are: the zero vector is in \(W\), for any scalar \(a\) and vector \(x\) in \(W\), \(ax \in W\), and for any vectors \(x\), \(y\) in \(W\), \(x+y \in W\).

Step by step solution

01

Define Subspaces

Recall that a subspace W of a vector space V is a subset of V, satisfying the following conditions: 1. The zero vector is in W. 2. For any scalar a and any vector x in W, ax is in W. 3. For any vectors x, y in W, x+y is in W.
02

Prove that W is a subspace if it satisfies the given conditions

We are given that W is nonempty and satisfies the following conditions: for any scalar a and any vectors x, y in W, ax is in W and x+y is in W. Let's show that W is a subspace by proving the three conditions of a subspace. 1. Since W is nonempty, it contains at least one vector. Let's call it v. When a=0, we get the zero vector as 0v, because \(0\boldsymbol{v} = \boldsymbol{0}\). So, the zero vector is in W. 2. For any scalar a and vector x in W, the given condition states that ax is in W. 3. Similarly, for any vectors x, y in W, the given condition states that x+y is in W. Since W satisfies all three subspace conditions, W is a subspace of V.
03

Prove that the given conditions hold true for a subspace

Now, let's assume that W is a subspace of V and show that W satisfies the conditions from the exercise. 1. According to the definition of a subspace, the zero vector is in W. Therefore, W is nonempty. 2. For any scalar a and vector x in W, ax is in W, as per the definition of a subspace. 3. For any vectors x, y in W, x+y is in W, also as per the definition of a subspace. Thus, whenever W is a subspace of V, it satisfies the given conditions from the exercise. #Conclusion# We have shown that a subset W of a vector space V is a subspace of V if and only if W satisfies the conditions of being nonempty and, whenever a is a scalar and x, y are vectors in W, then ax is in W and x+y is in W.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Subspace of a Vector Space
Understanding what makes a subset of a vector space a subspace is crucial for comprehending the structure and behavior of vector spaces in mathematical contexts like linear algebra. A subspace is more than just any collection of vectors; it鈥檚 a special set that retains the properties necessary to qualify as a vector space in its own right.

To dig deeper, here are the must-meet criteria for a subset to be a bona fide subspace:
  • The zero vector must be included in the subset. This provides an identity element for vector addition within the subspace.
  • Scalar multiplication must be closed within the subset. If you take any vector in the subset and multiply it by any scalar, the result should also be in the subset.
  • Vector addition must be closed within the subset. This means if you select any two vectors from the subset, their sum must also be part of the subset.
If these conditions are met, then algebraic operations within the subset will behave just like they do in the overall space鈥攈ence it's a 'sub' space. This allows us to treat subspaces as vector spaces for study and application.
Zero Vector
In any vector space, there is a unique vector, called the zero vector, that is the additive identity. Whenever you add the zero vector to any other vector, you simply get the original vector back.

The zero vector is of pivotal importance when dealing with subspaces. To be considered a subspace, the set must include this zero vector. It鈥檚 like saying that the subspace must have a starting point, a ground zero from which all linear combinations and operations extend. This aligns with our intuitive understanding of zero in arithmetic鈥攁 place of origin, a balance point, and a signifier of null quantity.
Scalar Multiplication
Scalar multiplication is one of the fundamental operations in a vector space. This operation involves taking a vector and a scalar (a number from the field over which the vector space is defined) and 'scaling' the vector by that number.

But why does it matter for subspaces? It's simple: if you multiply a vector in a subspace by a scalar, the result must also be in the subspace. If it's not, then we're dealing with a set that doesn't maintain the structure necessary for a vector space. Ensuring the closure under scalar multiplication means no matter how you stretch or shrink a vector in the subspace, you're still within the confines of that space.
Vector Addition
Vector addition is another cornerstone of vector spaces, allowing the combination of vectors to form new vectors. The rule for a subset to be a subspace ensures that the sum of any two vectors in the subset stays within the subset. This property is known as being 'closed under addition' and is just as important as the closure under scalar multiplication.

It's a non-negotiable condition. Without it, we can't perform all the vector space operations solely within the subset, thereby disqualifying it from being a subspace. The algebra of vectors in the subspace must be self-contained, echoing the behavior of the broader vector space it is part of.

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Most popular questions from this chapter

Let \(F\) be a field that is not of characteristic two. Define $$ \mathrm{W}_{1}=\left\\{A \in \mathrm{M}_{n \times n}(F): A_{i j}=0 \text { whenever } i \leq j\right\\} $$ and \(\mathrm{W}_{2}\) to be the set of all symmetric \(n \times n\) matrices with entries from \(F\). Both \(\mathrm{W}_{1}\) and \(\mathrm{W}_{2}\) are subspaces of \(\mathrm{M}_{n \times n}(F)\). Prove that \(\mathrm{M}_{n \times n}(F)=\mathrm{W}_{1} \oplus \mathrm{W}_{2}\). Compare this exercise with Exercise 28 .

Write the zero vector of \(M_{3 \times 4}(F)\).

Let \(V\) be a finite-dimensional vector space over \(C\) with dimension \(n\). Prove that if \(\mathrm{V}\) is now regarded as a vector space over \(R\), then \(\operatorname{dim} \mathrm{V}=\) \(2 n\). (See Examples 11 and 12.)

The vectors \(u_{1}=(1,1,1,1), u_{2}=(0,1,1,1), u_{3}=(0,0,1,1)\), and \(u_{4}=(0,0,0,1)\) form a basis for \(F^{4}\). Find the unique representation of an arbitrary vector \(\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\) in \(\mathrm{F}^{4}\) as a linear combination of \(u_{1}, u_{2}, u_{3}\), and \(u_{4}\).

Prove that the set of \(m \times n\) upper triangular matrices is a subspace of \(\mathrm{M}_{m \times n}(F)\).
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