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3. For each of the following sets, determine if it is a subspace of R3. If...
QUESTION 2. (a) Decide whether each of the following subsets of R’ is a subspace. Either...
QUESTION 2. (a) Decide whether each of the following subsets of R’ is a subspace. Either provide a proof showing the set is a subspace of R3, or provide a counterexample showing it is not a subspace: [9 marks] (i) S= {(x, y, z) ER3 : 4.0 + 9y + 8z = 0} (ii) S = {(x, y, z) E R3 : xy = 0} (b) Determine for which values of b ER, the set S = {(x, y, z)...
True or false: V = is a subspace of R3
True or false: $$ V=\left\{\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \in \mathbb{R}^{3}: x \geq 0\right\} $$is a subspace of R3. True False Question 10 (1 point) True or false: $$ V=\left\{\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \in \mathbb{R}^{3}: x-y=z+1\right\} $$is a subspace of R3. True False
Determine which of the following sets are subspaces of the given vector space. If it is...
Determine which of the following sets are subspaces of the given vector space. If it is NOT a subspace, circle NO and give a property that fails. Circle YES if it is a subspace, but you do NOT have to prove it. Let a and b be real numbers. a a+b (b) Let S= 1 . Is S a subspace of R3? YES or NO (c) Let S = {p(t) |p(t) = at + bt}. Is S a subspace of...
Are the following subsets subspaces of the given vector space?Justify your answers using words and proper...
Are the following subsets subspaces of the given vector space?Justify your answers using words and proper mathematical notation. If the set is not a subspace of the given vector space, give a counterex- ample (an example that demonstrates that one of the axioms fails) and explain why this shows the subset is not a subspace. If the set is a subspace, then prove it by showing that the conditions for a subset to be a subspace are met (a) S...
Determine whether each of the following is a subspace of the relevant R". (a) V1 =...
Determine whether each of the following is a subspace of the relevant R". (a) V1 = {(x, y, z) | x, y ER, Z E Z} (b) V2 = {(2,4,4) + s(1, 2, 2) + t(4,5,7) | ste R} (c) V3 = {(a, b, c, d) | a, b, c, d e R, ab = 0}
2. Find the closest point to y = in the subspace H = Span [ o།...
2. Find the closest point to y = in the subspace H = Span [ o། [ 17 [10] 3. Let B = {| 2 |,|-2， 1}. Find the coordinate vector of x = [1] relative to the [=1] [4] [2] orthogonal basis B for R3. ངོ- v1cs None of the above 5. Which of the following is true about the sets of vectors S and T? 3 1 [3 ] , 2 ), T={l U L-13] The set S...
Determine whether the following sets are linearly dependent or linearly indepen dent. If they are linearly...
Determine whether the following sets are linearly dependent or linearly indepen dent. If they are linearly dependent, find a subset that is linearly independent and has the same span (b) ((1,-1,2), (1,-2, 1), 1,4, 1)) in R3. (c) (1, 1,0), (1,0, 1), (0,1,1in (F2) (recall that F2-Z/2Z, the field with two elements).
1. Prove that each of the following is a subspace. (a) W = {x: x =...
1. Prove that each of the following is a subspace. (a) W = {x: x = (x 1, 22, 23) and X1 + 12 = x;} (b) W = {p: p(t) = ata + b + c and a+b+c=0} (C) W = {A € R2x2 and A is upper triangular) (d) W = {f:f EC(0,1) and f(0 =0} 2. Show that the following subsets of A R2x2 are not subspaces. (a) W = {A : A is the singular matrix}...
part a and b PROBLEM (HAND-IN ASSIGNMENT) Use the Subspace Test to determine whether the following...
part a and b
PROBLEM (HAND-IN ASSIGNMENT) Use the Subspace Test to determine whether the following sets W are subspaces of the given vector spaces: (A) The set W to be of all triples of real numbers (x, y, z) satisfying that 2x - 3y + 5z = 0 with the standard operations on Ris a subspace of R3. (B) The set of all 2 x 2 invertible matrices with the standard matrix addition and scalar multiplication.
Determine whether or not the following transformation T :V + W is a linear transformation. If...
Determine whether or not the following transformation T :V + W is a linear transformation. If T is not a linear transformation, provide a counter example. If it is, then: (i) find the nullspace N(T) and nullity of T, (ii) find the range R(T) and rank of T, (iii) determine if T is one-to-one, (iv) determine if T is onto. : (a) T: R3 + R2 defined by T(x, y, z) = (2x, y, z) (b) T: R2 + R2...
QUESTION 2. (a) Decide whether each of the following subsets of R’ is a subspace. Either provide a proof showing the set is a subspace of R3, or provide a counterexample showing it is not a subspace: [9 marks] (i) S= {(x, y, z) ER3 : 4.0 + 9y + 8z = 0} (ii) S = {(x, y, z) E R3 : xy = 0} (b) Determine for which values of b ER, the set S = {(x, y, z)...
True or false: $$ V=\left\{\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \in \mathbb{R}^{3}: x \geq 0\right\} $$is a subspace of R3. True False Question 10 (1 point) True or false: $$ V=\left\{\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \in \mathbb{R}^{3}: x-y=z+1\right\} $$is a subspace of R3. True False
Determine which of the following sets are subspaces of the given vector space. If it is NOT a subspace, circle NO and give a property that fails. Circle YES if it is a subspace, but you do NOT have to prove it. Let a and b be real numbers. a a+b (b) Let S= 1 . Is S a subspace of R3? YES or NO (c) Let S = {p(t) |p(t) = at + bt}. Is S a subspace of...
Are the following subsets subspaces of the given vector space?Justify your answers using words and proper mathematical notation. If the set is not a subspace of the given vector space, give a counterex- ample (an example that demonstrates that one of the axioms fails) and explain why this shows the subset is not a subspace. If the set is a subspace, then prove it by showing that the conditions for a subset to be a subspace are met (a) S...
Determine whether each of the following is a subspace of the relevant R". (a) V1 = {(x, y, z) | x, y ER, Z E Z} (b) V2 = {(2,4,4) + s(1, 2, 2) + t(4,5,7) | ste R} (c) V3 = {(a, b, c, d) | a, b, c, d e R, ab = 0}
2. Find the closest point to y = in the subspace H = Span [ o། [ 17 [10] 3. Let B = {| 2 |,|-2， 1}. Find the coordinate vector of x = [1] relative to the [=1] [4] [2] orthogonal basis B for R3. ངོ- v1cs None of the above 5. Which of the following is true about the sets of vectors S and T? 3 1 [3 ] , 2 ), T={l U L-13] The set S...
Determine whether the following sets are linearly dependent or linearly indepen dent. If they are linearly dependent, find a subset that is linearly independent and has the same span (b) ((1,-1,2), (1,-2, 1), 1,4, 1)) in R3. (c) (1, 1,0), (1,0, 1), (0,1,1in (F2) (recall that F2-Z/2Z, the field with two elements).
1. Prove that each of the following is a subspace. (a) W = {x: x = (x 1, 22, 23) and X1 + 12 = x;} (b) W = {p: p(t) = ata + b + c and a+b+c=0} (C) W = {A € R2x2 and A is upper triangular) (d) W = {f:f EC(0,1) and f(0 =0} 2. Show that the following subsets of A R2x2 are not subspaces. (a) W = {A : A is the singular matrix}...
part a and b
PROBLEM (HAND-IN ASSIGNMENT) Use the Subspace Test to determine whether the following sets W are subspaces of the given vector spaces: (A) The set W to be of all triples of real numbers (x, y, z) satisfying that 2x - 3y + 5z = 0 with the standard operations on Ris a subspace of R3. (B) The set of all 2 x 2 invertible matrices with the standard matrix addition and scalar multiplication.
Determine whether or not the following transformation T :V + W is a linear transformation. If T is not a linear transformation, provide a counter example. If it is, then: (i) find the nullspace N(T) and nullity of T, (ii) find the range R(T) and rank of T, (iii) determine if T is one-to-one, (iv) determine if T is onto. : (a) T: R3 + R2 defined by T(x, y, z) = (2x, y, z) (b) T: R2 + R2...
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