Question

Show that linear independence is preserved by one-to-one transformations and that spanning sets are preserved by onto transformations. More precisely, if \(T: V \rightarrow W\) is a linear transformation, show that: a. If \(T\) is one-to-one and \(\left\\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{n}\right\\}\) is independent in \(V,\) then \(\left\\{T\left(\mathbf{v}_{1}\right), \ldots, T\left(\mathbf{v}_{n}\right)\right\\}\) is independent in \(W\) b. If \(T\) is onto and \(V=\operatorname{span}\left\\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{n}\right\\},\) then \(W=\operatorname{span}\left\\{T\left(\mathbf{v}_{1}\right), \ldots, T\left(\mathbf{v}_{n}\right)\right\\}\)

Short answer

Linear independence and spanning sets are preserved under one-to-one and onto transformations, respectively.

Step-by-step solution

Understanding Linear Independence

A set of vectors \(\{\mathbf{v}_1, \ldots, \mathbf{v}_n\}\) in a vector space \(V\) is said to be linearly independent if the only scalars \(c_1, \ldots, c_n\) that satisfy the equation \(c_1 \mathbf{v}_1 + \ldots + c_n \mathbf{v}_n = \mathbf{0}\) are \(c_1 = c_2 = \ldots = c_n = 0\).

One-to-One Transformation Effect

A linear transformation \(T: V \rightarrow W\) is one-to-one (injective) if \(T(\mathbf{v}) = T(\mathbf{u})\) implies \(\mathbf{v} = \mathbf{u}\) for all \(\mathbf{v}, \mathbf{u} \in V\). This affects the preservation of linear independence because if a combination \(c_1 T(\mathbf{v}_1) + \ldots + c_n T(\mathbf{v}_n) = \mathbf{0}\) is possible without all \(c_i\) being zero, then the map is not one-to-one, contradicting the assumption.

Proving Preservation of Independence

Suppose \(c_1 T(\mathbf{v}_1) + \ldots + c_n T(\mathbf{v}_n) = \mathbf{0}\) with not all \(c_i = 0\). Then \(T(c_1 \mathbf{v}_1 + \ldots + c_n \mathbf{v}_n) = \mathbf{0}\). Since \(T\) is injective, it follows that \(c_1 \mathbf{v}_1 + \ldots + c_n \mathbf{v}_n = \mathbf{0}\), contradicting independence. Thus, \(c_1 = c_2 = \ldots = c_n = 0\).

Understanding Spanning Sets

A set of vectors \(\{\mathbf{v}_1, \ldots, \mathbf{v}_n\}\) spans a vector space \(V\) if every vector in \(V\) can be expressed as a linear combination of these vectors.

Onto Transformation Effect

A transformation \(T: V \rightarrow W\) is onto (surjective) if for every \(\mathbf{w} \in W\), there exists \(\mathbf{v} \in V\) such that \(T(\mathbf{v}) = \mathbf{w}\). This means every element of \(W\) is mapped from some element of \(V\), which implies any spanning set of \(V\) can span \(W\) through \(T\).

Proving Preservation of Spanning Sets

Since \(V\) is spanned by \(\{\mathbf{v}_1, \ldots, \mathbf{v}_n\}\), every \(\mathbf{v} \in V\) can be written as a linear combination of \(\{\mathbf{v}_1, \ldots, \mathbf{v}_n\}\). Because \(T\) is onto, for any \(\mathbf{w} \in W\), there is some \(\mathbf{v} \) in \(V\) such that \(T(\mathbf{v}) = \mathbf{w}\). Thus, \(W\) is spanned by \(T(\mathbf{v}_1), \ldots, T(\mathbf{v}_n)\).

Key concepts

Linear Independence

Linear independence is a crucial concept in linear algebra that tells us how vectors in a set interact with each other. To say that a set of vectors, \(\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}\), is linearly independent means that no vector in this set can be written as a linear combination of the others.
Practically, this means that the only way to combine these vectors to get the zero vector is by multiplying each vector by zero and adding up. Formally, \(c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \ldots + c_n \mathbf{v}_n = \mathbf{0}\) only when \(c_1 = c_2 = \ldots = c_n = 0\).
Understanding when vectors are independent helps us in tasks like checking if a system of equations has a unique solution or not. Methods for determining independence include using determinants and row reduction techniques.

Spanning Sets

Spanning sets refer to a collection of vectors whose linear combinations cover an entire vector space. If you have a set \(\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}\), they are said to span a vector space \(V\) if any vector in \(V\) can be expressed as a combination of these vectors.
In simpler terms, with a spanning set, you can "reach" any point in your vector space by scaling and adding the vectors in your set appropriately. Checking if a set of vectors spans a space involves ensuring that every vector in the space can be rewritten as a linear combination of the set.
Utilizing spanning sets is essential for understanding the structure of vector spaces and their subspaces. It's commonly used in applications where one needs to express certain directions or actions in terms of known vectors.

Injective and Surjective Functions

Functions in mathematics can map elements from one set to another in various ways, and understanding these mappings is critical in linear algebra. An injective function, or one-to-one function, ensures that every element in the original vector space \(V\) is mapped to a unique element in the target space \(W\). No two different elements in \(V\) will lead to the same output in \(W\). This property is essential when preserving linear independence during transformations.
On the other hand, a surjective function, or onto function, means the function covers every element of the target space \(W\). Every element in \(W\) has a pre-image in the original space \(V\). This feature is crucial for preservation of spanning sets as it means that the transformation reaches all parts of the new space.
In the context of linear transformations, checking if a function is injective or surjective helps in understanding the effects on vector independence or span.

Vector Spaces

Vector spaces form the backbone of linear algebra, encompassing sets of vectors that can be scaled and added together under certain rules. A vector space over a field (often real or complex numbers) includes vectors and follows properties like associativity, commutativity of addition, distributive properties, and existence of zero vector and additive inverses.
They can be finite-dimensional, like \(\mathbb{R}^n\), or infinite-dimensional. Understanding vector spaces involves getting to grips with concepts like basis, dimension, linear transformations, and subspaces.
Important questions about vector spaces often involve checking whether they have been defined properly using linearly independent or spanning sets. The structure serves as a foundation for many concepts both in pure and applied mathematics, making them critical for studies in areas like geometry, algebra, and functional analysis.