(a) Show that if L: V →V is linear and ker L ≠{0}, then L is not invertible. (b) Show that if im

step 1 All right, so follow me on this math, we have T of F of T equal to T times F of T. So given that

(a) Show that if L: V →V is linear and ker L ≠{0}, then L is...

(a) Show that if $L: V \rightarrow V$ is linear and $\operatorname{ker} L \neq\{0\}$, then $L$ is not invertible. (b) Show that if img $L \neq V$, then $L$ is not invertible. (c) Give an example of a linear map with $\operatorname{ker} L=\{0\}$ that is not invertible. Hint: First explain why your example must be on an infinite-dimensional vector space.

(a) Show that if $L: V \rightarrow V$ is linear and $\operatorname{ker} L \neq\{0\}$, then $L$ is not invertible.
(b) Show that if img $L \neq V$, then $L$ is not invertible.
(c) Give an example of a linear map with $\operatorname{ker} L=\{0\}$ that is not invertible. Hint: First explain why your example must be on an infinite-dimensional vector space.

Key Concepts

Linear Transformation

A linear transformation is a function between two vector spaces that preserves vector addition and scalar multiplication. These maps are fundamental in linear algebra as they provide a structured approach to study the behavior of vectors under different operations, leading to deep connections between algebraic structures and geometric interpretations.

Kernel

The kernel (or null space) of a linear map consists of all vectors that are mapped to the zero vector. It is a crucial concept because it indicates whether the linear transformation is one-to-one. A nontrivial kernel implies that the transformation fails to be injective, which is directly linked to its noninvertibility.

Image

The image (or range) of a linear transformation is the set of all output vectors that can be obtained from the input vectors in the domain. It is important in determining whether the map is surjective. When the image does not cover the entire target space, the linear map is not surjective and, hence, not invertible.

Invertibility of Linear Maps

A linear map is invertible if and only if it is both injective (one-to-one) and surjective (onto). This means that every element in the target space has a unique pre-image in the domain. Concepts of kernel and image are essential in verifying both of these properties, and failure in either leads to noninvertibility.

Finite-dimensional vs Infinite-dimensional Spaces

In finite-dimensional vector spaces, a linear map is invertible if it is either injective or surjective, as these properties imply each other. However, in infinite-dimensional spaces, a linear map can be injective without being surjective, leading to cases where a map with a trivial kernel is still not invertible. This distinction is fundamental when extending finite-dimensional intuitions to the infinite-dimensional context.

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