Linear Transform Synonyms: Grouping Components and Spaces

In summary, the conversation discussed various components of matrix multiplication, including the number of components in a vector and the dimensions of the resulting matrix. It also addressed the concept of a map from $\mathbb{R}^d$ to $\mathbb{R}^k$ and the different synonyms for describing the properties of a matrix, such as rank and nullity. The groupings of synonyms were also clarified.
  • #1
Kaspelek
26
0
Also just working on another question, especially stuck with the last part.

It's basically definitions.

View attachment 825

This is what I've got so far, correct me if I'm wrong.

a) k components, k components.
b) R^n to R^n
c) R^rank(T)
d)R^nullity(T)
e) Completely unsure (need help with this)
 

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  • #2
Kaspelek said:
Also just working on another question, especially stuck with the last part.

It's basically definitions.

View attachment 825

This is what I've got so far, correct me if I'm wrong.

A few things wrong over here, let's get started:

Kaspelek said:
a) k components, k components.

Remember how matrix multiplication works: the number of columns of the matrix on the left has to match the number of rows of the matrix on the right. In this case, A has d rows, which means that x (a column vector) has to have d components. The product, b, will have the same number of columns as x (one column) and the same number of rows as A (k rows). So, b has k components.

Kaspelek said:
b) R^n to R^n

I don't see an "n" anywhere in this question. Since A takes a vector of d components and gives a vector of k components, A is a map from $\mathbb{R}^d$ to $\mathbb{R}^k$.

Kaspelek said:
c) R^rank(T)
d)R^nullity(T)
e) Completely unsure (need help with this)

The answers to c) and d) are $\mathbb{R}^k$ and $\mathbb{R}^d$ respectively.

I'll put e) as it's own post.
 
  • #3
Now for e): Let's make a list for reference. For our purposes, I don't care about the difference between A and $T_A$ (which gives us a few answers for free, incidentally).

1. row space of A
2. kernel of A
3. column space of A
4. solution space of A
5. image of A
6. null space of A
7. rank of A
8. rank of A
9. dimension of image of A
10. nullity of A
11. nullity of A
12. dimension of the row space of A
13. dimension of the kernel of A
14. dimension of the image of A^T
15. nullity of A^T
16. number of rows of A
17. rank of A^T

The question is to "group all synonyms." Now I'm fairly sure the answer should be:
1
2,6
3,5
4
7,8,9,12,14,17
10,11,13
15
16

any questions on a particular grouping?
 

FAQ: Linear Transform Synonyms: Grouping Components and Spaces

What is a linear transform?

A linear transform is a mathematical operation that transforms a set of data points by applying a set of linear equations. It can also be referred to as a linear transformation or a linear map. It is commonly used in fields such as mathematics, physics, and engineering to analyze and manipulate data.

How is a linear transform different from other types of transforms?

A linear transform is different from other types of transforms in that it follows the rules of linearity. This means that the transformation preserves the properties of addition and scalar multiplication. In simpler terms, the output of a linear transform can be calculated by simply adding or multiplying the inputs by a constant factor.

What is the purpose of using a linear transform?

The purpose of using a linear transform is to simplify and analyze data in a more organized and manageable way. By transforming data using linear equations, it becomes easier to identify patterns, relationships, and trends in the data. It also allows for easier manipulation and calculation of the data.

Can a linear transform be applied to any type of data?

Yes, a linear transform can be applied to any type of data as long as the data can be represented in a mathematical form. This includes numerical data, such as values and measurements, as well as non-numerical data, such as images and text.

What are some real-world applications of linear transforms?

Linear transforms have various real-world applications, including image and signal processing, data compression, data analysis and visualization, and machine learning. They are also used in fields such as economics, biology, and statistics to analyze and interpret data.

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