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Find an ONB (orthonormal basis) for the following plane in R3 2 + y + 3z...
(i) Find an orthonormal basis {~u1, ~u2} for S (ii) Consider the function f : R3...
(i) Find an orthonormal basis {~u1, ~u2} for S
(ii) Consider the function f : R3 -> R3 that to each vector ~v
assigns the vector of S given by
f(~v) = <~u1, ~v>~u1 + <~u2; ~v>~u2. Show that it is a
linear function.
(iii) What is the matrix of f in the standard basis of R3?
(iv) What are the null space and the column space of the matrix
that you computed in the
previous point?
Exercise 1. In...
Find a basis for the following plane in R3 1 + y - 2z = 0...
Find a basis for the following plane in R3 1 + y - 2z = 0 First, solve the system, then assign parameters s and t to the free variables in this order), and write the solution in vector form as su + tv. Below, enter the components of the vectors u - (un, uz, uz)and v = (1, 0, vy)". ty and U-
Find an orthonormal basis for the subspace of R3 spanned by Extend the basis you found...
Find an orthonormal basis for the subspace of R3
spanned by
Extend the basis you found to an orthonormal basis for R 3 (by
adding a new vector or vectors). Is there a unique way to extend
the basis you found to an orthonormal basis of R3 ?
Explain.
1 -1.2 5 Uį = U2 = -3 1, U3 = 2 , 14 = 29...
1 -1.2 5 Uį = U2 = -3 1, U3 = 2 , 14 = 29 ( 7 Answer the following questions and give proper explanations. (a) Is {ui, U2, uz} a basis for R3? (b) Is {ui, U2, u4} a basis for R4? (c) Is {ui, U2, U3, U4, u; } a basis for R? (d) Is {ui, U2, U3, u} a basis for Rº?! (e) Are ui, u, and O linearly independent?! Problem 6. (15 points). Let A...
Please attempt both questions. 5. Find an orthonormal basis for the plane viewed as a subspace...
Please attempt both questions.
5. Find an orthonormal basis for the plane viewed as a subspace of R3. Z (-1,0,2) (0,-1,0) (0,1,0) X 6. Determine if each basis is orthogonal. Further, is the basis orthonormal? (a) In the vector space R3 (i.e. column vectors in 3-space): 1 2 5 -3 (b) In the vector space that consists of polynomial functions of degree less than or equal to 2: {f(x) = 22 - 3, 9() = 4, h(x) = 2² +2}...
(a) Write the vector aas a linear combination of the set of orthonormal basis vectors 2 marks] (b) Find the orthogonal...
(a) Write the vector aas a linear combination of the set of orthonormal basis vectors 2 marks] (b) Find the orthogonal projection of the vector (1,-3) on the vector v- (-1,5). 2 marks] (c) Using your result for part (b) verify that w = u-prolvu is perpendicular to V. 2 marks]
(a) Write the vector aas a linear combination of the set of orthonormal basis vectors 2 marks] (b) Find the orthogonal projection of the vector (1,-3) on the vector...
Use the following steps to find the general equation of the plane that intersects the surface...
Use the following steps to find the general equation of the plane that intersects the surface f(x, y) ye2x-y+5 at f(-1,3) Choose any vector ū, u # 0, in the xy-plane that is parallel to neither the x-axis nor the y а) axis. Use a cross product to show that u is parallel to neither axis. Find Duf-1,3) b) Choose any vector v, v *0, in the xy-plane that is orthogonal to u. Use the dot product to show that...
Use the following steps to find the general equation of the plane that intersects the surface...
Use the following steps to find the general equation of the plane that intersects the surface f (x, y) ye2x-y+5 at f(-1,3): = the y Choose any vector ū, u 0, in the xy-plane that is parallel to neither the x-axis nor product to show that i is parallel to neither axis. a) axis. Use a cross Find Df1,3) b) Choose any vector v, v 0, in the xy-plane that is orthogonal to u. Use the dot product to show...
Will rate once all is completed. 1) 2) 3) 4) (12 points) Find a basis of...
Will rate once all is completed.
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2)
3)
4)
(12 points) Find a basis of the subspace of R that consists of all vectors perpendicular to both El- 1 1 0 and 7 Basis: , then you would enter [1,2,3],[1,1,1] into the answer To enter a basis into WeBWork, place the entries. each vector inside of brackets, and enter a list these vectors, separated by commas. For instance if vour basis is 31 2 and u (12 points) Let...
ote: The norm of is denoted by |vand is calculated N a vector u Consider a subspace W of R4, W span(1, v2, v3, v4)). Where 0 из- 1. Find an orthonormal basis Qw of W and find the dimension of W 2. Fi...
ote: The norm of is denoted by |vand is calculated N a vector u Consider a subspace W of R4, W span(1, v2, v3, v4)). Where 0 из- 1. Find an orthonormal basis Qw of W and find the dimension of W 2. Find an orthonormal basis QWL of WL and find the dimension of WL 3. GIven a vector u- . find the Qw coordinate of Projw(v) . find the Qwa coordinate of Projwi (v) » find the coordinate...
(i) Find an orthonormal basis {~u1, ~u2} for S
(ii) Consider the function f : R3 -> R3 that to each vector ~v
assigns the vector of S given by
f(~v) = <~u1, ~v>~u1 + <~u2; ~v>~u2. Show that it is a
linear function.
(iii) What is the matrix of f in the standard basis of R3?
(iv) What are the null space and the column space of the matrix
that you computed in the
previous point?
Exercise 1. In...
Find a basis for the following plane in R3 1 + y - 2z = 0 First, solve the system, then assign parameters s and t to the free variables in this order), and write the solution in vector form as su + tv. Below, enter the components of the vectors u - (un, uz, uz)and v = (1, 0, vy)". ty and U-
Find an orthonormal basis for the subspace of R3
spanned by
Extend the basis you found to an orthonormal basis for R 3 (by
adding a new vector or vectors). Is there a unique way to extend
the basis you found to an orthonormal basis of R3 ?
Explain.
1 -1.2 5 Uį = U2 = -3 1, U3 = 2 , 14 = 29 ( 7 Answer the following questions and give proper explanations. (a) Is {ui, U2, uz} a basis for R3? (b) Is {ui, U2, u4} a basis for R4? (c) Is {ui, U2, U3, U4, u; } a basis for R? (d) Is {ui, U2, U3, u} a basis for Rº?! (e) Are ui, u, and O linearly independent?! Problem 6. (15 points). Let A...
Please attempt both questions.
5. Find an orthonormal basis for the plane viewed as a subspace of R3. Z (-1,0,2) (0,-1,0) (0,1,0) X 6. Determine if each basis is orthogonal. Further, is the basis orthonormal? (a) In the vector space R3 (i.e. column vectors in 3-space): 1 2 5 -3 (b) In the vector space that consists of polynomial functions of degree less than or equal to 2: {f(x) = 22 - 3, 9() = 4, h(x) = 2² +2}...
(a) Write the vector aas a linear combination of the set of orthonormal basis vectors 2 marks] (b) Find the orthogonal projection of the vector (1,-3) on the vector v- (-1,5). 2 marks] (c) Using your result for part (b) verify that w = u-prolvu is perpendicular to V. 2 marks]
(a) Write the vector aas a linear combination of the set of orthonormal basis vectors 2 marks] (b) Find the orthogonal projection of the vector (1,-3) on the vector...
Use the following steps to find the general equation of the plane that intersects the surface f(x, y) ye2x-y+5 at f(-1,3) Choose any vector ū, u # 0, in the xy-plane that is parallel to neither the x-axis nor the y а) axis. Use a cross product to show that u is parallel to neither axis. Find Duf-1,3) b) Choose any vector v, v *0, in the xy-plane that is orthogonal to u. Use the dot product to show that...
Use the following steps to find the general equation of the plane that intersects the surface f (x, y) ye2x-y+5 at f(-1,3): = the y Choose any vector ū, u 0, in the xy-plane that is parallel to neither the x-axis nor product to show that i is parallel to neither axis. a) axis. Use a cross Find Df1,3) b) Choose any vector v, v 0, in the xy-plane that is orthogonal to u. Use the dot product to show...
Will rate once all is completed.
1)
2)
3)
4)
(12 points) Find a basis of the subspace of R that consists of all vectors perpendicular to both El- 1 1 0 and 7 Basis: , then you would enter [1,2,3],[1,1,1] into the answer To enter a basis into WeBWork, place the entries. each vector inside of brackets, and enter a list these vectors, separated by commas. For instance if vour basis is 31 2 and u (12 points) Let...
ote: The norm of is denoted by |vand is calculated N a vector u Consider a subspace W of R4, W span(1, v2, v3, v4)). Where 0 из- 1. Find an orthonormal basis Qw of W and find the dimension of W 2. Find an orthonormal basis QWL of WL and find the dimension of WL 3. GIven a vector u- . find the Qw coordinate of Projw(v) . find the Qwa coordinate of Projwi (v) » find the coordinate...
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