Question

We write vectors ${\mathbb{R}}^n$ as rows. Suppose that $\mathbf{x},\mathbf{y},\mathbf{z},\mathbf{w}$ is a basis of ${\mathbb{R}}^4$. Show that: a. $\mathbf{x}+a\mathbf{w},\mathbf{y},\mathbf{z},\mathbf{w}$ is also a basis of ${\mathbb{R}}^4$ for any choice of the scalar $a$. b. $\mathbf{x}+\mathbf{w},\mathbf{y}+\mathbf{w},\mathbf{z}+\mathbf{w},\mathbf{w}$ is also a basis of ${\mathbb{R}}^4$. c. $\mathbf{x},\mathbf{x}+\mathbf{y},\mathbf{x}+\mathbf{y}+\mathbf{z},\mathbf{x}+\mathbf{y}+\mathbf{z}+\mathbf{w}$ is also a basis of ${\mathbb{R}}^4$

Short answer

All given sets are bases of ${\mathbb{R}}^4$, as shown by proving their linear independence and spanning.

Step-by-step solution

Understand the Concept of Basis

A basis is a set of vectors in a vector space where the vectors are linearly independent, and they span the entire vector space. Since {$\mathbf{x},\mathbf{y},\mathbf{z},\mathbf{w}$} forms a basis of ${\mathbb{R}}^4$, these vectors are linearly independent and span ${\mathbb{R}}^4$.

Show Linear Independence for Part (a)

To prove that {$\mathbf{x}+a\mathbf{w},\mathbf{y},\mathbf{z},\mathbf{w}$} is a basis, we must show these vectors are linearly independent. Assume a linear combination results in the zero vector: $(c_1(\mathbf{x}+a\mathbf{w})+c_2\mathbf{y}+c_3\mathbf{z}+c_4\mathbf{w}=\mathbf{0})$. This simplifies to $c_1\mathbf{x}+c_2\mathbf{y}+c_3\mathbf{z}+(c_{1}a+c_4)\mathbf{w}=\mathbf{0}$. Since $\mathbf{x},\mathbf{y},\mathbf{z},\mathbf{w}$ are a basis of ${\mathbb{R}}^4$, we conclude all coefficients are zero: $c_1=c_2=c_3=c_4=0$. Thus, the vectors are independent.

Linear Independence Implies Spanning for Part (a)

Since the vectors in {$\mathbf{x}+a\mathbf{w},\mathbf{y},\mathbf{z},\mathbf{w}$} are linearly independent and there are four of them, they must span ${\mathbb{R}}^4$ as well. Hence, this set is a basis.

Show Linear Independence for Part (b)

Consider the set {$\mathbf{x}+\mathbf{w},\mathbf{y}+\mathbf{w},\mathbf{z}+\mathbf{w},\mathbf{w}$}. Assume a linear combination results in zero: $c_1(\mathbf{x}+\mathbf{w})+c_2(\mathbf{y}+\mathbf{w})+c_3(\mathbf{z}+\mathbf{w})+c_4\mathbf{w}=\mathbf{0}$. This groups to $c_1\mathbf{x}+c_2\mathbf{y}+c_3\mathbf{z}+(c_1+c_2+c_3+c_4)\mathbf{w}=\mathbf{0}$. As the original set forms a basis, the coefficients must again be zero, $c_1=c_2=c_3=c_4=0$.

Linear Independence Implies Spanning for Part (b)

Since {$\mathbf{x}+\mathbf{w},\mathbf{y}+\mathbf{w},\mathbf{z}+\mathbf{w},\mathbf{w}$} is linearly independent and contains four vectors, it spans ${\mathbb{R}}^4$. This set is therefore a basis.

Show Linear Independence for Part (c)

Consider the set {$\mathbf{x},\mathbf{x}+\mathbf{y},\mathbf{x}+\mathbf{y}+\mathbf{z},\mathbf{x}+\mathbf{y}+\mathbf{z}+\mathbf{w}$}. Assume a linear combination leads to the zero vector: $c_1\mathbf{x}+c_2(\mathbf{x}+\mathbf{y})+c_3(\mathbf{x}+\mathbf{y}+\mathbf{z})+c_4(\mathbf{x}+\mathbf{y}+\mathbf{z}+\mathbf{w})=\mathbf{0}$. Simplifying gives $(c_1+c_2+c_3+c_4)\mathbf{x}+(c_2+c_3+c_4)\mathbf{y}+(c_3+c_4)\mathbf{z}+c_4\mathbf{w}=\mathbf{0}$. For a basis, all individual coefficients are zero, showing $c_1=0,c_2=0,c_3=0,c_4=0$.

Linear Independence Implies Spanning for Part (c)

Similarly, since {$\mathbf{x},\mathbf{x}+\mathbf{y},\mathbf{x}+\mathbf{y}+\mathbf{z},\mathbf{x}+\mathbf{y}+\mathbf{z}+\mathbf{w}$} consists of four linearly independent vectors, it spans ${\mathbb{R}}^4$. This set is thereby a basis.

Key concepts

Vector Space

In Linear Algebra, a vector space is a fundamental concept. It is a set of elements called vectors, where two operations, vector addition and scalar multiplication, are defined and satisfy certain rules. For instance, the set of all two-dimensional vectors forms a vector space.

Vector spaces can be exemplified in many forms including spaces like ${\mathbb{R}}^n$, which consists of all n-tuples of real numbers. Every vector in ${\mathbb{R}}^n$ can be visualized as a point or position in n-dimensional space. The space ${\mathbb{R}}^4$, for example, contains vectors with four components.

• A vector space must satisfy properties like commutativity and associativity of addition.
• There must exist a zero vector that acts as an additive identity.
• Every vector must have an inverse and must distribute over scalar multiplication.

These properties ensure that vector spaces provide a structured way to handle linear transformations and equations.

Basis

The concept of a basis in a vector space is crucial because it provides a foundation for representing every vector in the space. A basis is comprised of vectors that are both linearly independent and which span the entire vector space.

In ${\mathbb{R}}^4$, if you have a set of four vectors, say $$\{\mathbf{x},\mathbf{y},\mathbf{z},\mathbf{w}\}$$, they form a basis if no vector can be written as a combination of the others.

• Linear independence ensures there are no redundant vectors in the set.
• Spanning means you can construct any vector in the space by combining basis vectors.

Knowing the basis enables simpler and often more efficient computation and analysis of vectors within the space. For example, manipulating the vectors of a basis allows you to generate new bases that might be more convenient for certain tasks, without losing the essential characteristic of spanning and independence.

Linear Independence

Linear independence is a property of a set of vectors that indicates the vectors do not overlap in directions or dimensions in the vector space. In simpler terms, no vector in the set can be made by adding up multiples of the others.

This property is crucial for a set to be considered a basis of a vector space. Suppose you have vectors $({\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n)$. They are linearly independent if the only solution to $c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2+\cdots +c_n{\mathbf{v}}_n=\mathbf{0}$ is when all coefficients $c_1,c_2,\dots ,c_n$ are zero.

• This means each vector adds a new dimension or direction.
• Without linear independence, you could swap one vector for a multiple of another, making it unnecessary to describe the space fully.

Verifying linear independence involves solving systems of equations in linear algebra, emphasizing its fundamental role in understanding and manipulating vector spaces.

Span

In vector spaces, the span of a set of vectors is the collection of all possible vectors you can create using linear combinations of those vectors. It's like stretching out a canvas to cover every possible combination you can achieve with that initial set.

For example, in the vector space ${\mathbb{R}}^4$, a set of vectors spans the space if you can form any vector in ${\mathbb{R}}^4$ by multiplying and adding the vectors in your set. To determine if vectors span a space, you assess whether their combinations can fill the goal space.

• Spanning is essential for a basis; without it, the vectors wouldn't adequately describe the entire space.
• Understanding span involves using matrices and solving equations to ensure full coverage of dimensions.

Essentially, when a set of vectors spans a space, it means you've got all angles and reaches covered, which ensures flexibility in representing data within that space.