Linear Algebra : Operations and Properties
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Example Question #501 : Operations And Properties
There is a relationship between the dimension of the domain and the dimension of the null space and the rank of a function. The relationship is
where 
and 
Let 
6
3
0
2
8
2
The answer is 
The dimension of the domain is 
The rank of f is 
Plug these into the equation given above to get
Therefore, the dimension of the null space is 
Example Question #502 : Operations And Properties
There is a relationship between the dimension of the domain and the dimension of the null space and the rank of a function. This question is on that.
Let 
Use the formula
where 
and 
From the problem statement, we know 
Example Question #503 : Operations And Properties
Let 
The answer is 
Use the formula
where 
and 
From the problem statement, we know 
Example Question #21 : Linear Mapping
Let 
not enough information
no
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 
Yes
Not enough information
No
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 
No
Not enough information
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 #504 : 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 
Not enough information
No
Yes
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)
Not enough information
Yes
No
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 
Not enough information
No
Yes
Yes
The linear map, 
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