Linear Algebra : Linear Mapping
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Example Questions
Example Question #502 : Operations And Properties
Let 
no
not enough information
yes
no
The rank and the dimension of the null space must add up to the dimension of the domain from the formula
where 
and 
For this problem the dimension of the domain (
From the problem statement, we want to consider 
Hence the equation for dimension doesn't hold.
Example Question #21 : Linear Mapping
A mapping is said to be onto (sometimes called surjective) if it's image is the entire codomain.
Is the linear map 
Not enough information
No
Yes
Yes
Yes, 
Example Question #22 : Linear Mapping
A mapping is said to be onto (sometimes called surjective) if it's image is the entire codomain.
Is the linear map 
(This map is sometimes called a projection, specifically a projection onto the 
Not enough information
No
Yes
Yes
This mapping is onto. For any given vector in the codomain, 
Example Question #23 : Linear Mapping
Is the linear map 
(Note this is sometimes called the identity map because it maps every vector to itself)
Not enough information
No
Yes
Yes
The answer is yes, the mapping is onto. The image of 
Example Question #506 : Operations And Properties
A mapping is said to be onto (sometimes called surjective) if it's image is the entire codomain.
Is the linear map 
No
Yes
Not enough information
No
First we need to find the image of 
Treat 
The image of 
Another way to do this problem is recognize that the domain,
Example Question #24 : Linear Mapping
A mapping is said to be onto (sometimes called surjective) if it's image is the entire codomain.
Is the linear map 
(Note this is called the zero mapping)
No
Yes
Not enough information
No
No, 
Example Question #25 : Linear Mapping
A mapping is said to 1-to-1 (sometimes called injective) if no two vectors in the domain go to the same vector in the image of the mapping.
Is the linear map 
Yes
Not enough information
No
Yes
The linear map, 
Example Question #26 : Linear Mapping
There are relationships between the dimension of the null space (sometimes called kernal) and if a function is 1-to-1
The dimension of a linear map's null space is zero. Is the linear map 1-to-1?
Yes
No
Not enough information
Yes
A linear map is always 1-to-1 if the null space has dimension zero.
The converse of this statement is also true. A linear map is 1-to-1 if the null space has dimension 0.
Example Question #26 : Linear Mapping
A mapping is said to 1-to-1 (sometimes called injective) if no two vectors in the domain go to the same vector in the image of the mapping.
Is the linear map
(Note this is called the zero mapping)
No
Not enough information
Yes
No
The linear map, 
For example, the vector 
Example Question #27 : Linear Mapping
A linear map 
Not enough information
No
Yes
No
This linear mapping is not 1-to-1. We know this because the null space is spanned by two vectors. Therefore the null space has dimension 
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