Consider the transformation T[x y] = [x + y y^2]
a. Is T a linear transformation?
b. Is the range of T closed under addition?
c. "" scalar multiplication?
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Consider the transformation T[x y] = [x + y y^2]
a. Is T a linear transformation?...
linear algebra do both drawing in part A and B 2. Consider the set and let...
linear algebra
do both drawing in part A and B
2. Consider the set and let addition and scalar multiplication be the standard operations on vectors in R2 (a) State a specific numerical example that shows that V is not closed under vector addition, then draw a sketch for your example, shading the region V, and drawing your example vectors and their linear combination with the parallelogram method that shows V is not closed under vector addition (b) Provide a...
1. Consider the transformation given by T(x, y, z)- (2z 3z+) (a) Show that T is...
1. Consider the transformation given by T(x, y, z)- (2z 3z+) (a) Show that T is a linear transformation (b) Find the domain and range of T (c) Find the number of columns and r for T. (d) Find the standard matrix for T.
20. Consider the transformation from R →Rdefined by T(x, y, z) = (x + y, z)....
20. Consider the transformation from R →Rdefined by T(x, y, z) = (x + y, z). a. Under this transformation, find the image of the ordered pair (1, -3, 2). b. Is the transformation linear? Show your work! [5 marks]
= Let T:R3 → Rº be the linear transformation given by T(x,y,z) = (x – 2,...
= Let T:R3 → Rº be the linear transformation given by T(x,y,z) = (x – 2, x + y, x + y + 2z) for all (x,y,z) e R3. Determine whether T is invertible or not. If T is invertible, find the inverse of T and compute inverse image of (1,1,1) under T.
Consider the function T: K3 K3 defined by T(x, y, z) = (0, y,0). This kind...
Consider the function T: K3 K3 defined by T(x, y, z) = (0, y,0). This kind of function is called a projection, since we are 'projecting' the vector (2, y, z) onto the y-axis. In this problem, you will prove that the function T is linear. In the first part, you will prove that T preserves addition. In the second part, you will prove that T preserves scalar multiplication. There is only one correct answer for each part, so be...
2. [& marks] Consider the line ar transformation T: R – R? T(x,y,z) = (x +y-2,...
2. [& marks] Consider the line ar transformation T: R – R? T(x,y,z) = (x +y-2, -1-y+z). (a) Show that the matrix [T]s, representing T in the standard bases of Rand R' is of the form [7|6,6= ( +1 -1 1). -1 -1 1 (b) Find a basis of the null space of T and determine the dimension of this space. (c) Find a basis of the range of T and determine the dimension of the range of T. (d)...
2. (8 marks] Consider the linear transformation T:R3 R2 TX,Y, 2) = (+y-2, -1-y+z). (a) Show...
2. (8 marks] Consider the linear transformation T:R3 R2 TX,Y, 2) = (+y-2, -1-y+z). (a) Show that the matrix (TE.Es representing T in the standard bases of R3 and R² is of the form TEE 1 -1 1 -1 -1 1 (b) Find a basis of the null space of T and determine the dimension of this space. (c) Find a basis of the range of T and determine the dimension of the range of T. (d) Is T Onto?...
:{f(x) € C(-1,1): f(x) = f(-x)} a subspace of C(-1, 1)? Yes No, it is closed...
:{f(x) € C(-1,1): f(x) = f(-x)} a subspace of C(-1, 1)? Yes No, it is closed under scalar multiplication but not addition No, it is closed under addition but not scalar multiplication No, it is neither closed under either addition nor 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...
Find the standard matrix for the linear transformation T. T(x, y, z) = (x + y,...
Find the standard matrix for the linear transformation T. T(x, y, z) = (x + y, X- (x + y, X – 2, 2 – x) III III ul.
linear algebra
do both drawing in part A and B
2. Consider the set and let addition and scalar multiplication be the standard operations on vectors in R2 (a) State a specific numerical example that shows that V is not closed under vector addition, then draw a sketch for your example, shading the region V, and drawing your example vectors and their linear combination with the parallelogram method that shows V is not closed under vector addition (b) Provide a...
1. Consider the transformation given by T(x, y, z)- (2z 3z+) (a) Show that T is a linear transformation (b) Find the domain and range of T (c) Find the number of columns and r for T. (d) Find the standard matrix for T.
20. Consider the transformation from R →Rdefined by T(x, y, z) = (x + y, z). a. Under this transformation, find the image of the ordered pair (1, -3, 2). b. Is the transformation linear? Show your work! [5 marks]
= Let T:R3 → Rº be the linear transformation given by T(x,y,z) = (x – 2, x + y, x + y + 2z) for all (x,y,z) e R3. Determine whether T is invertible or not. If T is invertible, find the inverse of T and compute inverse image of (1,1,1) under T.
Consider the function T: K3 K3 defined by T(x, y, z) = (0, y,0). This kind of function is called a projection, since we are 'projecting' the vector (2, y, z) onto the y-axis. In this problem, you will prove that the function T is linear. In the first part, you will prove that T preserves addition. In the second part, you will prove that T preserves scalar multiplication. There is only one correct answer for each part, so be...
2. [& marks] Consider the line ar transformation T: R – R? T(x,y,z) = (x +y-2, -1-y+z). (a) Show that the matrix [T]s, representing T in the standard bases of Rand R' is of the form [7|6,6= ( +1 -1 1). -1 -1 1 (b) Find a basis of the null space of T and determine the dimension of this space. (c) Find a basis of the range of T and determine the dimension of the range of T. (d)...
2. (8 marks] Consider the linear transformation T:R3 R2 TX,Y, 2) = (+y-2, -1-y+z). (a) Show that the matrix (TE.Es representing T in the standard bases of R3 and R² is of the form TEE 1 -1 1 -1 -1 1 (b) Find a basis of the null space of T and determine the dimension of this space. (c) Find a basis of the range of T and determine the dimension of the range of T. (d) Is T Onto?...
:{f(x) € C(-1,1): f(x) = f(-x)} a subspace of C(-1, 1)? Yes No, it is closed under scalar multiplication but not addition No, it is closed under addition but not scalar multiplication No, it is neither closed under either addition nor 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...
Find the standard matrix for the linear transformation T. T(x, y, z) = (x + y, X- (x + y, X – 2, 2 – x) III III ul.
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