3. In R4 , does the set {(1, 1, 1, 0,(1, 0, 0, 0),(0, 1, 0,...

3. In R⁴ , does the set {(1, 1, 1, 0,(1, 0, 0, 0),(0, 1, 0, 0),(0, 0, 1, 1)}, span R⁴? In other words, can you write down any vector (a, b, c, d) ∈ R⁴ as a linear combination of vectors in the given set ? Is the above set of vectors linearly independent ?

4. In the vector space P₂ of polynomials of degree ≤ 2, find explicitly a polynomial p(x) which is not in the span of the set {x + 2, x² − 1}.

5. Let S be the subspace of P₂ defined by S := {ax² + bx + 2a + 3b : a, b ∈ R}, for different choices of real numbers a and b (you don’t need to show here that S is indeed a subspace, and can assume. But is a good practice problem). Find a basis, and hence dimension for S.

Expert Solution

Colby Messinger answered 2 years ago

Find a basis for R4 that contains the vectors X = (1, 2, 0, 3)⊤ and...

Find a basis for R4 that contains the vectors X = (1, 2, 0, 3)⊤ and ⊤ Y =(1,−3,5,10)T.

Consider the set of vectors S = {(1, 0, 1),(1, 1, 0),(0, 1, 1)}. (a) Does...

Consider the set of vectors S = {(1, 0, 1),(1, 1, 0),(0, 1, 1)}. (a) Does the set S span R3? (b) If possible, write the vector (3, 1, 2) as a linear combination of the vectors in S. If not possible, explain why.

Let S be the set of vectors in R4 S ={(1,1,-1,2),(1,0,1,3),(1,-3,2,2),(0,-2-1,-2)} (a) State whether the vectors...

Let S be the set of vectors in R4 S ={(1,1,-1,2),(1,0,1,3),(1,-3,2,2),(0,-2-1,-2)} (a) State whether the vectors in the set are linearly independent. (b) Find the basis for S. (c) Find the dimension of the column space (rank S). (d) Find the dimension of the null space (nullity S). (e) Find the basis for the null space of S?

A linear transformation from R3-R4 with the V set of vectors x, where T(x)=0, is V...

A linear transformation from R3-R4 with the V set of vectors x, where T(x)=0, is V a subspace of R3?

Consider the set of integer numbers from 0 to 9, that is {0, 1, 2, 3,...

Consider the set of integer numbers from 0 to 9, that is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Bob wishes to use these numbers to create a 7-digit password to secure his new laptop. Note that each number can appear in any position (for example, 0 can be the first number in the password). (a) Find the number of 7-digit passwords that are possible. (b) Find the number of 7-digit passwords with distinct digits. (c) Find...

1. With mutation set to 0, migration set to 0, all genotypes with equal fitness, and...

1. With mutation set to 0, migration set to 0, all genotypes with equal fitness, and assortative mating set to 0, what would you expect to happen and why? a. allele frequencies should change dramatically, because microevolution is occurring b. new alleles will be entering the population, which will change allele frequencies c. some phenotypes are more adaptive than others, so they will increase in frequency over generations d. allele frequencies should stay about the same, because microevolution is not...

Consider the set of vectors {(?,?,?) ∈ ?3???ℎ ?ℎ?? ? − 3? + 5? = 0.}...

Consider the set of vectors {(?,?,?) ∈ ?3???ℎ ?ℎ?? ? − 3? + 5? = 0.} Does this set of vectors form a sub-space of ?3?

(6) In each part below, find a basis for R4 CONTAINED IN the given set, or...

(6) In each part below, find a basis for R4 CONTAINED IN the given set, or explain why that is not possible: Do not use dimension (a) {(1,1,0,0),(1,0,1,0),(0,1,1,0)} (b) {(1,−1,0,0),(1,0,−1,0),(0,1,−1,0),(0,0,0,1)} (c) {(1,1,0,0),(1,−1,0,0),(0,1,−1,0),(0,0,1,−1)} (d) {(1,1,1,1),(1,2,3,4),(1,4,9,16),(1,8,27,64),(1,16,81,256)}

(5) In each part below, find a basis for R4 that contains the given set, or...

(5) In each part below, find a basis for R4 that contains the given set, or explain why that is not possible: (a) {(1,1,0,0),(1,0,1,0),(0,1,1,0)} (b) {(1,−1,0,0),(1,0,−1,0),(0,1,−1,0)} (c) {(1,1,0,0),(1,−1,0,0),(0,1,−1,0),(0,0,1,−1)} (d) {(1,1,1,1),(1,2,3,4),(1,4,9,16),(1,8,27,64),(1,16,81,256)}

eigenvalues of the matrix A = [1 3 0, 3 ?2 ?1, 0 ?1 1] are...

eigenvalues of the matrix A = [1 3 0, 3 ?2 ?1, 0 ?1 1] are 1, ?4 and 3. express the equation of the surface x^2 ? 2y^2 + z^2 + 6xy ? 2yz = 16. How should i determine the order of the coefficient in the form X^2/A+Y^2/B+Z^2/C=1?

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