Find the standard matrix for the linear operator T: R^3 --> R^3 that first rotates a vector counterclockwise about the z-axis through an angle “90" degrees, then reflects the resulting vector

Please write down the 2D rotation matrix that rotates by 90 degrees clockwise as a product of three shear matrices

#3

Assume that T is a linear transformation. Find the standard matrix of T. 7x T: R² →R² first rotates points through radians and then reflects points through the horizontal x₁-axis. A= (Type an integer or simplified fraction for each matrix element. Type exact answers, using radicals as needed.)

Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify your answer. T: R³-R², T(e₁) = (1,3), T(e₂) = (2, -11), and T(e3)= (-3,8), where e₁,e₂, e3 are the columns of the 3x3 identity matrix. a. Is the linear transformation one-to-one? A. T is not one-to-one because the standard matrix A has a free variable. B. T is one-to-one because the column vectors are not scalar multiples of each other. O C. T is not one-to-one because the columns of the standard matrix A are linearly independent. O D. T is one-to-one because T(x) = 0 has only the trivial solution. b. Is the linear transformation onto? O A. T is not onto because the standard matrix A contains a row of zeros. O B. T is onto because the standard matrix A does not have a pivot position for every row. O C. T is not onto because the columns of the standard matrix A span R². O D. T is onto because the columns of the standard matrix A span R².

Let's create a matrix that reflects around a line that is perpendicular to the vector n = [0.375, 0.927] in 2D, this line passes though the origin. Now that you have both rows, what is the image of the point (1, 2) under this transformation? Enter the x-coordinate, round it to 3 decimal places.

Determine whether the span of the column vectors of the given matrix A is equal to R^3. Show your work and justify your answer. 1 1 1 A = 1 2 0 1 0 1 |

Use matrix multiplication to find the orthogonal projection of (-8, 5, 7) on the xy -plane. Let T denote the linear operator mapping each vector into its orthogonal projection on the xy -plane. Then T(-8,5,7) = ( Ma )

8) Determine the ranks of the coefficient matrix A and the augmented matrix A^b. 9) Solve the simultaneous equations,and express the result in the form of a vector X.

Put as many 1's as possible in a 4 by 7 echelon matrix U whose pivot columns are(a) 2, 4, 5                (b)1,3,6,7 (c) 4 and 6.