# Image and Kernel

TLDRThe video explains the concepts of image and kernel in linear algebra. It defines image as the set of vectors obtained when transforming a subspace of the vector space V into the vector space W under a linear transformation. Kernel is defined as the set of vectors in V that get mapped to the zero vector in W under the transformation. An example transformation from R3 to R2 finding its image and kernel on certain subspaces is discussed. Properties of image and kernel as subspaces are noted. The goal is to provide viewers a conceptual overview of these foundational linear algebra ideas.

###### Takeaways

- π Image refers to the set of vectors we get when transforming a subspace of the domain
- π Kernel refers to vectors in the domain that map to the zero vector in the codomain
- π Image of a subspace is a subspace of the codomain
- π Kernel of a transformation is a subspace of the domain
- π To find the image, transform a subset of the domain and see where in the codomain it maps to
- π To find the kernel, solve the equation L(v) = 0 for vectors v in the domain
- π€ Example shows mapping R3 to R2, image of a subspace is vectors of form (c,2c)
- π Example shows kernel of the mapping are vectors in R3 of form (0,c,c)
- π§ Properties of subspaces apply to kernels and images
- π€ Definitions and examples help explain the concepts of image and kernel

###### Q & A

### What are image and kernel in linear algebra?

-Image and kernel are two related concepts in linear algebra. The image of a subset S of vector space V under a linear transformation is the set of vectors in vector space W that you get when you apply the transformation to all vectors in S. The kernel of a linear transformation is the set of vectors in V that get mapped to the zero vector in W.

### How do you denote the image of a vector space V under transformation L?

-The image of the entire vector space V under linear transformation L is denoted as Im(L) or R(L). This set is also known as the range of L.

### What is an intuitive way to think about the image of a subset S under a linear transformation?

-You can think of the image as shining a light on the vectors in S and seeing which vectors in W light up when you apply the transformation. The vectors that light up form the image of S.

### How do you find the kernel of a linear transformation?

-To find the kernel, you set the transformation of a general vector v equal to the zero vector in the target space. Then solve for the vectors v that satisfy this equation. The set of v vectors that map to the zero vector is the kernel.

### What are some key properties of image and kernel?

-The image of any subset S is a subspace of the target vector space W. The kernel is always a subspace of the original vector space V. Vectors in either the image or kernel follow properties of subspaces like closure under addition and scalar multiplication.

### What was the specific linear transformation provided as an example?

-The example transformation was from R3 to R2, defined by L(v) = (v1, v2 - v3) for any v in R3. This takes a 3D vector and maps it to a 2D vector.

### What was the subspace S used to demonstrate image in the example?

-The subspace S was vectors in R3 of the form (c, 2c, 0) where c is a scalar. These are vectors where the 2nd element is twice the 1st and the 3rd element is 0.

### What was the image of S under the example transformation L?

-The image was vectors in R2 of the form (c, 2c) where the 2nd element is twice the 1st. Applying L to vectors in S produced this set of vectors.

### What was the kernel of the example transformation L?

- The kernel was vectors in R3 of the form (0, c, c) where the 1st element is 0 and 2nd and 3rd elements are equal. These vectors mapped to the zero vector in R2 under L.

### Why are image and kernel important concepts in linear algebra?

-Image and kernel help reveal key properties of a linear transformation. They show which vectors get mapped to zeros and what the range of the mapping is. This provides insight into the behavior and structure of the transformation.

###### Outlines

##### π Defining image and kernel

This paragraph defines the concepts of image and kernel in linear transformations. The image of a subspace S is the set of vectors obtained by transforming vectors from S. It can be visualized as the area of the target vector space that 'lights up' when transforming vectors from S. The kernel is the set of vectors that get mapped to the zero vector.

###### Mindmap

###### Keywords

##### π‘Linear transformation

##### π‘Image

##### π‘Range

##### π‘Kernel

##### π‘Subspace

##### π‘Vector spaces

##### π‘Mapping

##### π‘Function

##### π‘Vectors

##### π‘Zero vector

###### Highlights

Presents a new deep learning model for semi-supervised audio classification

Achieves state-of-the-art results on AudioSet using only 10% labeled data

Proposes a self-supervised pre-training method using spectrogram reconstruction

Pre-training helps model generalize to unseen classes using limited labels

Model combines CNN feature extractor, transformer encoder, and MLP classifier

Transformer encoder allows modeling long-range dependencies in audio

Spectrogram reconstruction pre-training is unsupervised and adaptable

Significantly outperforms fully supervised model when labels limited

Provides insights into semi-supervised learning for audio tasks

Could enable practical audio classification with minimal labeling

Limitations include computational cost and dataset dependence

Future work could explore different self-supervised pre-training tasks

Overall an important step towards semi-supervised audio processing

Code and models available in GitHub repo for reproducibility

Opens promising research direction in low-resource audio modeling

###### Transcripts

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