QuestionLet , and .
a. Is in ? How many vectors are in ?
b. How many vectors are in ?
c. Is in the subspace spanned by ? Why?
a. Is in ?
A. Vector is not in because it is not a linear combination of , and .
B. Vector is in because the subspace generated by , and is .
C. Vector is not in because it is not , or .
D. Vector is in because it is a linear combination of , and .
How many vectors are in ? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The number of vectors in is .
B. There are infinitely many vectors in .
b. How many vectors are in ? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The number of vectors in is .
B. There are infinitely many vectors in Span .
Studdy Solution
STEP 1
1. The set consists of three specific vectors.
2. The span of a set of vectors is the set of all possible linear combinations of those vectors.
3. To determine if a vector is in the span of other vectors, we need to check if it can be written as a linear combination of those vectors.
STEP 2
1. Determine if is in .
2. Count the number of vectors in .
3. Determine the number of vectors in .
4. Check if is in .
STEP 3
Determine if is in :
To check if is in the set , we need to see if is exactly equal to any of the vectors , , or .
Comparing with each vector:
-
-
-
is not equal to any of these vectors.
STEP 4
Count the number of vectors in :
The set contains exactly three vectors.
STEP 5
Determine the number of vectors in :
The span of a set of vectors is the set of all possible linear combinations of those vectors. Since are vectors in , the span can potentially be the entire space if they are linearly independent.
In general, the span of three vectors in can contain infinitely many vectors, as any linear combination of the vectors is possible.
STEP 6
Check if is in :
To determine if is in the span, we need to check if there exist scalars such that:
This translates to solving the following system of equations:
Solving this system will determine if such scalars exist.
STEP 7
Solve the system of equations:
Using substitution or elimination methods, solve the system:
1. From the second equation:
2. Substitute into the first and third equations:
- First equation:
- Simplify:
-
- Third equation:
- Simplify:
-
3. Solve the simplified system:
-
-
Adding these equations gives , which is consistent but does not determine unique values for .
Thus, can be expressed as a linear combination of .
The answers to the questions are:
a. C. Vector is not in because it is not , or .
A. The number of vectors in is .
b. B. There are infinitely many vectors in .
c. is in the subspace spanned by because it can be expressed as a linear combination of these vectors.
Was this helpful?