Joshua has $34 in his bank account. With his new job, he deposits the same amount each week into the account. After 35 weeks, Jo
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Answer:
1. Yes.
2. No.
3. Yes.
Step-by-step explanation:
Consider the following subsets of Pn given by
1.Let W1 be the set of all polynomials of the form , where a is in ℝ.
2.Let W2 be the set of all polynomials of the form , where a is in ℝ.
3. Let W3 be the set of all polynomials of the form , where a is in ℝ.
Recall that given a vector space V, a subset W of V is a subspace if the following criteria hold:
- The 0 vector of V is in W.
- Given v,w in W then v+w is in W.
- Given v in W and a a real number, then av is in W.
So, for us to check if the three subsets are a subset of Pn, we must check the three criteria.
- First property:
Note that for W2, for any value of a, the polynomial we get is not the zero polynomial. Hence the first criteria is not met. Then, W2 is not a subspace of Pn.
For W1 and W3, note that if a= 0, then we have p(t) =0, so the zero polynomial is in W1 and W3.
- Second property:
W1. Consider two elements in W1, say, consider a,b different non-zero real numbers and consider the polynomials
.
We must check that p_1+p_2(t) is in W1.
Note that
Since a+b is another real number, we have that p1(t)+p2(t) is in W1.
W3. Consider two elements in W3. Say . Then
So, again, p1(t)+p2(t) is in W3.
- Third property.
W1. Consider an element in W1 and a real scalar b. Then
.
Since (ba) is another real scalar, we have that bp(t) is in W1.
W3. Consider an element in W3 and a real scalar b. Then
.
Since (ba) is another real scalar, we have that bp(t) is in W3.
After all,
W1 and W3 are subspaces of Pn for n= 2
and W2 is not a subspace of Pn.
Answer: There are 6 sections of of rope Mario can cut from the rope .
Step-by-step explanation:
Since we have given that
Length of the rope for the tent = 7 feet
As we know that
Length of rope is used to tie a tarp on top of the tent = 34.5 inches
Remaining length of rope is given by
According to question, the remaining rope should be cut into
So, Number of sections of rope Mario can cut from the rope is given by
Hence, there are 6 sections of rope Mario can cut from the rope .