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11- Let A be the line xix2 in the x1x2-plane. A is a subspace of R2....
6. Let P be the subspace in R 3 defined by the plane x − 2y...
6. Let P be the subspace in R 3 defined by the plane x − 2y + z
= 0. (a) [5 points] Use the Gram–Schmidt process to find orthogonal
vectors that form a basis for P. (b) [5 points] Find the projection
p of b = (3, −6, 9) onto P.
6. Let P be the subspace in R3 defined by the plan 2y+z0 (a) [5 points] Use the Gram-Schmidt process to find orthogonal vectors that form a basis...
Linear Algebra Problem! 1. Let U be the subspace of R3 given by 11 + 12...
Linear Algebra Problem!
1. Let U be the subspace of R3 given by 11 + 12 - 213 = 0. for U. Justify that is an ordered basis. What is the a) Find an ordered basis dimension of U? b) Let ū= (1,1,1). Show that ✓ EU and find the B-coordinate vector (Ū3 = C:(Ū), where Ce: U + R2 is the B-coordinate transformation.
Let H = Let H= 1 33x2+59'51) 5y 51), which represents the set of points on...
Let H = Let H= 1 33x2+59'51) 5y 51), which represents the set of points on and inside an ellipse in the xy-plane. Find two specific examples scalar to show that H is not a subspace of R2. H is not a subspace of R2 because the two vectors 3 1 show that H is not closed under addition. (Use a comma to separate vectors as needed.) H is not a subspace of R2 because the scalar 4 and the...
Let H ⊂ R2 be the line H = {(a, 2a) : a ∈ R}. Consider the collection of all translates of H, i.e...
Let H ⊂ R2 be the line H = {(a, 2a) : a ∈ R}. Consider the collection of all translates of H, i.e., all lines in the plane with slope 2. Find the equivalence relation on R2 defined by this partition of R2 .
(Select ALL that are TRUE) R2 is a subspace of R3. Let (v1, ..., Un) be...
(Select ALL that are TRUE) R2 is a subspace of R3. Let (v1, ..., Un) be a basis for R" and A € Mnxn(R) be invertible. Then (Av1, ..., Avn) is a basis for R". Let (v, w, u) be a tuple in Rº. Then (v +w,v+u, u +w) is linearly independent.
1 point) Let H be the set of all points in the fourth quadrant in the plane V R2. That is, H- t(x...
1 point) Let H be the set of all points in the fourth quadrant in the plane V R2. That is, H- t(x, y) |z 2 0,y S 0. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? H contains the zero vector of V 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in H whose sum is not in H,...
Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let P2 R2 R2 be the orthogonal projection on...
Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let P2 R2 R2 be the orthogonal projection onto the line L2: x32 2r2 0. 0. (1) What are the image and kernel of P2P What is the rank of P2P? Give a geometric description, without relying on the matrix of P2P (2) Find the matrix that represents P2P
Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let...
The Moulton Plane is the plane M = (R2, LM) such that a subset I of...
The Moulton Plane is the plane M = (R2, LM) such that a subset I of R2 belongs to LM if and only if one of the following holds: i) l = {(x,y)| x=a} (vertical line); ii) l = {(x,y)| y=b} (horizontal line) iii) ( = {(x,y)| y = mx +b where m<0} (line with negative slope) [ m(x - x0) if x xo when m>0}. (bent line W 14,9 m ( x - x0) if x > xo with...
1 point) Let V R2 and let H be the subset of V of all points...
1 point) Let V R2 and let H be the subset of V of all points on the line-4x-3y-0. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? | H does not contain the zero vector of V | 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in H whose sum is not in H, using a comma separated list and...
13 State the closed graph theorem and use it to prove the following. Let H be a closed subspace o...
13 State the closed graph theorem and use it to prove the following. Let H be a closed subspace of C[0, 1] which is also closed in L,10, 1 (in L1-norm). It is known that the mapping I : h E H C CO, 1] → h E L1 is bounded. Show that 1-1 is bounded (continuous)
13 State the closed graph theorem and use it to prove the following. Let H be a closed subspace of C[0, 1] which...
6. Let P be the subspace in R 3 defined by the plane x − 2y + z
= 0. (a) [5 points] Use the Gram–Schmidt process to find orthogonal
vectors that form a basis for P. (b) [5 points] Find the projection
p of b = (3, −6, 9) onto P.
6. Let P be the subspace in R3 defined by the plan 2y+z0 (a) [5 points] Use the Gram-Schmidt process to find orthogonal vectors that form a basis...
Linear Algebra Problem!
1. Let U be the subspace of R3 given by 11 + 12 - 213 = 0. for U. Justify that is an ordered basis. What is the a) Find an ordered basis dimension of U? b) Let ū= (1,1,1). Show that ✓ EU and find the B-coordinate vector (Ū3 = C:(Ū), where Ce: U + R2 is the B-coordinate transformation.
Let H = Let H= 1 33x2+59'51) 5y 51), which represents the set of points on and inside an ellipse in the xy-plane. Find two specific examples scalar to show that H is not a subspace of R2. H is not a subspace of R2 because the two vectors 3 1 show that H is not closed under addition. (Use a comma to separate vectors as needed.) H is not a subspace of R2 because the scalar 4 and the...
(Select ALL that are TRUE) R2 is a subspace of R3. Let (v1, ..., Un) be a basis for R" and A € Mnxn(R) be invertible. Then (Av1, ..., Avn) is a basis for R". Let (v, w, u) be a tuple in Rº. Then (v +w,v+u, u +w) is linearly independent.
1 point) Let H be the set of all points in the fourth quadrant in the plane V R2. That is, H- t(x, y) |z 2 0,y S 0. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? H contains the zero vector of V 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in H whose sum is not in H,...
Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let P2 R2 R2 be the orthogonal projection onto the line L2: x32 2r2 0. 0. (1) What are the image and kernel of P2P What is the rank of P2P? Give a geometric description, without relying on the matrix of P2P (2) Find the matrix that represents P2P
Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let...
The Moulton Plane is the plane M = (R2, LM) such that a subset I of R2 belongs to LM if and only if one of the following holds: i) l = {(x,y)| x=a} (vertical line); ii) l = {(x,y)| y=b} (horizontal line) iii) ( = {(x,y)| y = mx +b where m<0} (line with negative slope) [ m(x - x0) if x xo when m>0}. (bent line W 14,9 m ( x - x0) if x > xo with...
1 point) Let V R2 and let H be the subset of V of all points on the line-4x-3y-0. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? | H does not contain the zero vector of V | 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in H whose sum is not in H, using a comma separated list and...
13 State the closed graph theorem and use it to prove the following. Let H be a closed subspace of C[0, 1] which is also closed in L,10, 1 (in L1-norm). It is known that the mapping I : h E H C CO, 1] → h E L1 is bounded. Show that 1-1 is bounded (continuous)
13 State the closed graph theorem and use it to prove the following. Let H be a closed subspace of C[0, 1] which...
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