exam 2 290

What are all of the possible spans for a pair of vectors in two-dimensional space? (Select all that apply.) What are all of the possible spans for a set of three vectors in three-dimension space? (Select all that apply.)

1: a single point, a line, a plane 2: a single point, a line, a plane, all three-dimensional space

Determine whether the set S is linearly independent or linearly dependent.S = {(6, 9), (-18, -27)} STEP 1:Determine if (-18, -27) is a scalar multiple of (6, 9). STEP 2:Determine if the set is linearly dependent.

1: scalar multiple 2: linearly dependent

Determine whether the subset of Mn,n is a subspace of Mn,n with the standard operations of matrix addition and scalar multiplication. The set of all n × n invertible matrices

not a subspace

Find all subsets of the set S = {(1, 0), (0, 1), (−1, −1)} that form a basis for R2. (Select all that apply.)

{(0, 1), (−1, −1)} {(1, 0), (−1, −1)} {(1, 0), (0, 1)}

Let V be the set of all positive real numbers. Determine whether V is a vector space with the following operations. x + y = xyAdditioncx = xcScalar multiplication If it is, then verify each vector space axiom; if it is not, then state all vector space axioms that fail. STEP 1:Check each of the 10 axioms.(1)-(10) = this axiom holds STEP 2:Use your results from Step 1 to decide whether V is a vector space.

(1) (2) All=This Axiom holds (3) (4) (5) (6) (7) (8) (9) (10) Step 2: V is a vector Space

Describe the additive inverse of a vector, (v1, v2, v3), in the vector space. R^3

(−v1, −v2, −v3)

Determine the dimension of the vector space. R4

4

Determine the dimension of the vector space. M2,3 STEP 1:Determine the number of linearly independent vectors needed to span M2,3.The basis for M2,3 has 6 linearly independent vectors. STEP 2:Using the result from Step 1, determine the dimension of M2,3.6

6 6 bc 2*3

Explain why S is not a basis for R2. S = {(−5, 7)}

S does not span R2.

Determine whether the set S spans R2. If the set does not span R2, then give a geometric description of the subspace that it does span. S = {(1, 6), (−2, −12), (4, 24)}

S does not span R2. S spans a line in R2.

Determine whether the set S spans R2. If the set does not span R2, then give a geometric description of the subspace that it does span. S = {(−3, 2)}

S does not span R2. S spans a line in R2.

Determine whether the set S spans R3. If the set does not span R3, then give a geometric description of the subspace that it does span. S = {(1, 0, 3), (2, 0, −1), (4, 0, 5), (2, 0, 6)}

S does not span R3. S spans a plane in R3.

Determine whether the set S spans R3. If the set does not span R3, then give a geometric description of the subspace that it does span. S = {(−2, 6, 0), (4, 7, 1)}

S does not span R3. S spans a plane in R3.

Determine whether S is a basis for R3. S = {(4, 3, 5), (0, 3, 5), (0, 0, 5)} If S is a basis for R3, then write u = (12, 6, 25) as a linear combination of the vectors in S. (Use s1, s2, and s3, respectively, as the vectors in S. If not possible, enter IMPOSSIBLE.)

S is a basis for R3. u=3s1-s2+3s3

Explain why S is not a basis for M2,2. S = [00;11] [01;10] [01;0−1] [10;00]

S is linearly dependent and does not span M2,2.

Explain why S is not a basis for R2. S = {(−7, 8), (0, 0)}

S is linearly dependent and does not span R2.

Explain why S is not a basis for P2. S = {1, 7x, −5 + x2, 8x}

S is linearly dependent.

Explain why S is not a basis for R2. S = {(2, 6), (1, 0), (0, 1)}

S is linearly dependent.

Explain why S is not a basis for R3. S = {(1, 1, 1), (0,1,1), (1,1,0), (0, 0, 0)}

S is linearly dependent.

Determine whether S is a basis for P3. S = {−2 + t3, 3t2, 2 + t, 2 + 2t + 3t2 + t3}

S is not a basis of P3.

Determine whether S is a basis for the indicated vector space. S = {(0, 0, 0), (6, 2, 1), (1, 3, 6)} for R3

S is not a basis of R3.

Determine whether S is a basis for the indicated vector space. S = {(0, 4, −1), (4, 0, 3), (−8, 20, −11)} for R3

S is not a basis of R3.

Determine whether the set S spans R2. If the set does not span R2, then give a geometric description of the subspace that it does span. S = {(2, −1), (−2, 2)}

S spans R2.

Determine whether the set S spans R3. If the set does not span R3, then give a geometric description of the subspace that it does span. S = {(5, 8, 2), (−3, 2, 6), (1, −4, 4)}

S spans R3.

Determine whether the set {v1, v2} is a basis for R^2.

The set is a basis of R^2.

Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify one of the vector space axioms that fails. The set of all 3 × 3 diagonal matrices with the standard operations

The set is a vector space.

Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify one of the vector space axioms that fails. The set of all 3 × 3 matrices of the form[000;abc;def] with the standard operations

The set is a vector space.

Determine whether the set {v1, v2} is a basis for R^2.

The set is not a basis of R2.

Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify one of the vector space axioms that fails. The set of all 3 × 3 nonsingular matrices with the standard operations

The set is not a vector space because it is not closed under addition.

Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify one of the vector space axioms that fails. The set of all first-degree polynomial functions ax, a ≠ 0, whose graphs pass through the origin with the standard operations

The set is not a vector space because it is not closed under addition.

Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify one of the vector space axioms that fails. The set of all fourth-degree polynomials with the standard operations

The set is not a vector space because it is not closed under addition.

Is W is a subspace of V? If not, state why. Assume that V has the standard operations. (Select all that apply.) W = {(x1, x2, x3, 0): x1, x2, and x3 are real numbers} V = R4

W is a subspace of V.

Is W a subspace of the vector space? If not, state why. (Select all that apply.) W is the set of all vectors in R2 whose second component is the cube of the first.

W is not a subspace of R2 because it is not closed under addition. W is not a subspace of R2 because it is not closed under scalar multiplication.

Is W a subspace of the vector space? If not, state why. (Select all that apply.) W is the set of all vectors in R2 whose components are integers.

W is not a subspace of R2 because it is not closed under scalar multiplication.

Is W a subspace of the vector space? If not, state why. (Select all that apply.) W is the set of all vectors in R2 whose components are rational numbers.

W is not a subspace of R2 because it is not closed under scalar multiplication.

Determine whether the set W is a subspace of R3 with the standard operations. If not, state why. (Select all that apply.) W = {(x1, 1/x1, x3): x1 and x3 are real numbers, x1 ≠ 0}

W is not a subspace of R3 because it is not closed under addition. W is not a subspace of R3 because it is not closed under scalar multiplication.

Determine whether the set W is a subspace of R3 with the standard operations. If not, state why. (Select all that apply.) W = {(x1, x2, 8): x1 and x2 are real numbers}

W is not a subspace of R3 because it is not closed under addition. W is not a subspace of R3 because it is not closed under scalar multiplication.

Is W a subspace of the vector space? If not, state why. (Select all that apply.) W is the set of all vectors in R3 whose second component is −3.

W is not a subspace of R3 because it is not closed under addition. W is not a subspace of R3 because it is not closed under scalar multiplication.

Is W a subspace of the vector space? W is the set of all matrices in Mn,n with zero determinants.

W is not a subspace of the vector space.

Consider the following. W = {(4t, t, −t): t is a real number} (a) Give a geometric description of the subspace W of R3. (b) Find a basis for the subspace W of R3. (c) Determine the dimension of the subspace W of R3

a line 4, 1, -1 1

For which values of t is each set linearly independent? (a) S = {(t, 1, 1), (1, t, 1), (1, 1, t)} (b) S = {(t, 1, 1), (1, 0, 1), (1, 1, 3t)}

a= All t ≠ 1, −2 b= All t ≠ 1/2

Rather than use the standard definitions of addition and scalar multiplication in R3, suppose these two operations are defined as follows. With these new definitions, is R3 a vector space? Justify your answers. (a) (x1, y1, z1) + (x2, y2, z2) = (x1 + x2, y1 + y2, z1 + z2)c(x, y, z) = (0, cy, cz) (b) (x1, y1, z1) + (x2, y2, z2) = (0, 0, 0)c(x, y, z) = (cx, cy, cz) (c) (x1, y1, z1) + (x2, y2, z2) = (x1 + x2 + 8, y1 + y2 + 8, z1 + z2 + 8)c(x, y, z) = (cx, cy, cz) (d) (x1, y1, z1) + (x2, y2, z2) = (x1 + x2 + 1, y1 + y2 + 1, z1 + z2 + 1)c(x, y, z) = (cx + c − 1, cy + c − 1, cz + c − 1)

a= The set is not a vector space because the multiplicative identity property is not satisfied. b= The set is not a vector space because the additive identity property is not satisfied. c= The set is not a vector space because the distributive property is not satisfied. d= The set is a vector space.

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) To show that a set is not a vector space, it is sufficient to show that just one axiom is not satisfied. (b) The set of all first-degree polynomials with the standard operations is a vector space. (c) The set of all pairs of real numbers of the form (0, y), with the standard operations on R^2, is a vector space.

a= True, for a set with two operations to be a vector space all 10 axioms must be satisfied. Therefore, if just one of the axioms fails, then this set cannot be a vector space. b= False. The set is not a vector space because it is not closed under addition. c= True. The set is a vector space because all 10 axioms are satisfied.

By inspection, determine if each of the sets is linearly dependent. (a) S = {(2, −1), (1, 3), (−4, 2)} (b) S = {(3, −6, 4), (9, −18, 12)} (c) S = {(0, 0), (2, 0)}

a= linearly dependent b= linearly dependent c= linearly dependent

Complete the proof of the remaining property of this theorem by supplying the justification for each step. Use the properties of vector addition and scalar multiplication from this theorem. (a) c−1(cv) = c−10 (b) c−1(cv) = 0 (c) (c−1c)v = 0 (d) 1v = 0 (e) v = 0

a= multiply both sides of the equation by a non-zero constant b= apply the property a0 = 0 for any scalar a c= apply the associative property of multiplication d= the product of multiplicative inverses is 1 e= apply the multiplicative identity property

Let {v1, v2, , vk} be a linearly independent set of vectors in a vector space V. Delete the vector vk from this set and prove that the set {v1, v2, , vk − 1} cannot span V. If the set {v1, v2, , vk−1} spanned V, then vk = c1v1 + c2v2 + + ck − 1vk − 1 for scalars c1, , ck−1 equal to some value . So, c1v1 + c2v2 + + ck − 1vk − 1 − vk = 0 which is impossible because {v1, v2, , vk} is linearly independent. Thus the set {v1, v2, , vk − 1} cannot span V.

equal to some value impossible

Determine whether the set S is linearly independent or linearly dependent. S = {(7, 0, 0), (0, 8, 0), (0, 0, −8), (7, 5, −4)}

linearly dependent

Determine whether the set of vectors in P2 is linearly independent or linearly dependent. S = {−2 − x, 2 + 2x + x2, 16 + 9x + x2}

linearly dependent

Determine whether the set S is linearly independent or linearly dependent. S = {(3, 1), (5, 4)}

linearly independent

Determine whether the set S is linearly independent or linearly dependent. S = {(−3, 1, 3), (1, 9, −2), (1, 4, −4)}

linearly independent

Determine whether the set of vectors in P2 is linearly independent or linearly dependent. S = {x2, 7 + x2}

linearly independent

Which describes the effect of multiplying a vector by a scalar less than −1.

stretching the vector and flipping it over the origin

Determine whether the set W is a subspace of R3 with the standard operations. W = {(x1, x2, 0): x1 and x2 are real numbers}

subspace

Determine whether the subset of Mn,n is a subspace of Mn,n with the standard operations of matrix addition and scalar multiplication. The set of all n × n lower triangular matrices

subspace